To Ukraine, they are a chit to play in an ongoing appeal to President Trump for more financial and military support. To Mr. Trump, they should be overdue payment for billions of dollars committed to Kyiv’s war effort.

Either way, Ukraine’s vast and valuable mineral resources have suddenly become a prominent component in the maneuvering over the country’s future.

Over the past week, Mr. Trump has repeatedly pushed the idea of trading U.S. aid for Ukraine’s critical minerals. He told Fox News on Monday that he wanted “the equivalent of like $500 billion worth of rare earths,” a group of minerals crucial for many high-tech products, in exchange for American aid. Ukraine had “essentially agreed to do that,” he said.

For Ukraine, it is a hopeful sign that Mr. Trump, a longtime skeptic of American aid to Kyiv, might find a path to maintaining support that he finds palatable. But it’s still possible that the famously mercurial president will change his mind, and even his statements about a deal have been ambiguous about whether he wants Ukraine’s minerals for past or future aid — or a combination of both.

Mr. Trump’s proposal followed a campaign launched by Kyiv in the fall to appeal to the U.S. president’s business-oriented mind-set by discussing lucrative energy deals and emphasizing that defending Ukraine aligned with American economic interests.

The campaign included a meeting between Mr. Trump and President Volodymyr Zelensky and trips to the United States by Ukrainian officials to pitch deals for exploiting deposits of lithium and titanium — vital for producing technologies like electric batteries. It also involved getting backing from influential political figures like the Republican senator Lindsey Graham.

The campaign was launched after politically connected U.S. investors started showing interest in Ukraine’s underground wealth in late 2023, despite the war that has been raging since 2022. A consortium including TechMet, an energy investment firm partly owned by the U.S. government, and Ronald S. Lauder, a wealthy friend of Mr. Trump, has engaged with Kyiv to bid on a Ukrainian lithium field, according to a letter to Mr. Zelensky reviewed by The New York Times.

Mr. Lauder, a cosmetics heir who planted the idea in Mr. Trump’s mind of buying resource-rich Greenland, said through a spokesman that he had not discussed Ukrainian minerals with Mr. Trump directly, but had “raised the issue with stakeholders in the U.S. and Ukraine for many years up to the present day.”

As Mr. Trump pushes for peace talks between Russia and Ukraine, Kyiv’s campaign around critical minerals has underscored Mr. Zelensky’s evolving strategy for retaining American support. Moving away from the moral appeals he used with the Biden administration, he has embraced a more transactional approach closer to Mr. Trump’s style. Mr. Zelensky recently said that he would also be interested in purchasing American liquefied natural gas.

Speaking to Reuters on Friday, Mr. Zelensky said: “If we are talking about a deal, then let’s do a deal.”

Mr. Trump said Tuesday that he was sending his Treasury secretary, Scott Bessent, to meet Mr. Zelensky in Ukraine — the first visit by a Trump administration official.

Mr. Bessent was directed to explore with the Ukrainians what a deal for the minerals might look like. He was expected in Kyiv on Wednesday, according to an official with knowledge of the trip.

The idea of leveraging Ukraine’s mineral resources took shape last summer. Kyiv was crafting its “Victory Plan,” a strategy aimed at ending the war on terms favorable to Ukraine, and wanted to convince its allies to sustain their support despite rising war fatigue.

Ukrainian officials were particularly concerned about a potential return to power of Mr. Trump, who had vowed to cut off military and financial aid to Ukraine, according to Ukrainian officials and lawmakers.

Their attention was drawn to an argument voiced by a Trump ally: Mr. Graham, who was making the case that Ukraine was “sitting on a gold mine” of critical minerals. “If we help Ukraine now, they could become the best business partner we’ve ever dreamed of,” he said last June. He reinforced his message in a video with Mr. Zelensky in September.

Ukrainian authorities say the country holds deposits of nearly half the 50 minerals the United States has identified as critical for its economy and national security. The Kyiv School of Economics says Ukraine holds the largest titanium reserves in Europe and a third of the continent’s lithium reserves.

Groups like SecDev, a Canadian firm, have valued the reserves at several trillion dollars. But some deposits are inaccessible because they are in Russian-occupied land. And experts caution that exploiting mineral reserves could be a costly and prolonged process, with new surveys needed to accurately assess their potential.

In a November 2023 letter to Mr. Zelensky, Brian Menell, head of TechMet, sought assistance for launching a bid to exploit a state-owned lithium field in central Ukraine. The letter named Mr. Lauder, the Texas-based investment firm Privateer Capital and other U.S. and international investors as associates.

Mr. Menell and other energy executives met Mr. Zelensky in New York in September. It is unclear whether they discussed the bid, which has yet to be launched. “TechMet, together with our partners, is available to move forward with further work if the U.S. and Ukrainian governments instruct us to do so,” Mr. Menell said in a statement.

Kyiv then decided to offer allies access to critical minerals as part of its Victory Plan. Mr. Zelensky presented the plan to Mr. Trump during a meeting in New York in late September.

During Mr. Trump’s first term, when Kremlin-backed troops were already waging war in eastern Ukraine, Kyiv won support from him, including for weapons supplies, by buying coal from an important swing state, Pennsylvania.

Part of Ukraine’s argument now is to emphasize that should it lose the war, its mineral resources would fall into the hands of Russia and its allies like China, which already dominates the global market for such materials.

John E. Herbst, a former U.S. ambassador to Ukraine and now senior director of the Atlantic Council’s Eurasia Center, said Ukraine’s pitch was an obvious sell for Mr. Trump.

“It involves vast energy resources, economically valuable assets which are not abundant in the Western world, and it’s a way to strip other competitors, including China, of their leverage over the United States,” Mr. Herbst said. “It’s a no-brainer.”

Last fall, work was underway to finalize and sign an agreement with the Biden administration to cooperate on extracting and processing minerals. But Ukrainian authorities decided to postpone the signing of the deal and offer it to Mr. Trump instead so that he could claim credit for it.

A Ukrainian delegation led by Yulia Svyrydenko, the economy minister, then traveled to New York and Washington in December for meetings with U.S. officials and business representatives, with critical minerals as a central topic.

Ms. Svyrydenko outlined potential energy deals, including the acquisition of production licenses for critical minerals. A presentation, seen by The Times, emphasized that Ukraine was “capable of building the entire value chain to meet U.S. and E.U. metallic titanium demand for 25 years.”

Matthew Murray, chairman of the advisory board of Velta, a Ukrainian titanium extraction company, said Ms. Svyrydenko had asked him and other U.S. business representatives for their help in making Ukraine’s case to the Trump administration.

“We spent a good deal of this meeting talking about critical raw materials,” Mr. Murray said. He added that the exploitation of these resources could become a new pillar of the U.S.-Ukraine relationship.

What shape this pillar may take remains to be seen.

A draft of the agreement that Kyiv postponed, and which was reviewed by The Times, included only pledges to share information and expertise on potential partnerships. It contained no financial commitments and was nonbinding. It is unclear whether Kyiv and Washington will amend the agreement to align it with Mr. Trump’s latest proposals.

Mr. Murray, a former Obama administration official, said one idea circulating in Washington was to offer Ukraine a loan to purchase U.S. weaponry, with access to critical minerals as collateral. This proposal could align with Mr. Trump’s vision, who has talked of obtaining a “guarantee” for continued American assistance.

“There are many steps yet to be taken, but the concept is very viable,” he said.

Marian Prysiazhniuk contributed reporting.

The leaders of the world’s two rival superpowers have plenty to talk about. But so far, it seems, they can’t agree on when they should speak, or even whether they already have.

Over the last 10 days or so, President Trump has sent conflicting messages about talking to China’s top leader, Xi Jinping. Mr. Trump said on Feb. 3 that they would have a call within “24 hours,” then reversed himself, saying he was in “no rush” to talk. More recently, when asked whether the two had spoken since he took office, Mr. Trump said that they had.

By contrast, Mr. Xi has said nothing about talking to Mr. Trump. And on Tuesday, China’s foreign ministry suggested that no such call had taken place since Mr. Trump’s return to the White House, apparently contradicting Mr. Trump’s claim, which he made during an interview with the Fox News host Bret Baier that aired this week.

Mr. Xi, it seems, is in no rush to engage, stalling progress on a number of thorny issues in the nations’ competition for global power. (Both sides say the leaders did speak on Jan. 17, before Mr. Trump’s inauguration.) Even as Mr. Trump has imposed tariffs aimed directly and indirectly at China, Mr. Xi has played it cool, preferring to be seen hosting foreign dignitaries at the opening of the Asian Winter Games in the icy northeastern Chinese city of Harbin.

The disconnect reflects, in part, how Mr. Xi is trying to show Mr. Trump and the Chinese people that he will not be intimidated by tariffs, analysts say. Mr. Xi does not want to be confused with Prime Minister Justin Trudeau of Canada or President Claudia Sheinbaum of Mexico, both of whom quickly negotiated with Mr. Trump after he threatened to slap 25 percent tariffs on their nations’ exports to the United States.

Those leaders “caved to Trump and Trump claimed victory,” said Zhiqun Zhu, an expert in Chinese foreign policy at Bucknell University in Pennsylvania. “Xi will not let that happen. It would reflect very, very badly on him domestically. I think the strategy is to wait and see what Trump can offer before taking a call.”

Mr. Zhu said Mr. Xi could also be irked by a joint statement that Mr. Trump and Japan’s prime minister, Shigeru Ishiba, released after their meeting in Washington last week, which addressed Taiwan, the self-governing island claimed by Beijing.

Mr. Trump and Mr. Ishiba said they opposed any changes to the status quo in the Taiwan Strait made not only by force, but by coercion. The mention of coercion, believed to be a first, refers to China’s use of nonmilitary means to pressure Taiwan, such as cyberattacks and trade restrictions.

To strike any kind of deal with Mr. Xi, Mr. Trump will almost certainly be pressed to soften Washington’s stance on Taiwan. That could include a pledge to reduce American arms sales to the island, or a statement that the United States “opposes” formal independence for Taiwan (currently, it says it “does not support” that).

It is unclear whether the Trump administration would even consider such concessions, which could be widely viewed as a blow to Taiwan’s security.

The administration may also be reluctant to yield on another major demand by Mr. Xi: loosening U.S. restrictions on exports of technology to China, such as the advanced semiconductors needed to power artificial intelligence, a key battleground between the world’s top two economic powers.

As for Mr. Trump, he wants China to reduce its trade surplus with the United States and do more to crack down on undocumented immigration, as well as the sale of chemicals used to make fentanyl. He has called for China to approve a sale of TikTok so that an American company can buy half of the video-sharing platform, and he has suggested that Beijing help his administration end the war in Ukraine.

Chinese analysts said that any promise from Mr. Trump would have to be viewed skeptically. Relations between the two countries seemed stable when Mr. Trump visited Beijing in November 2017 and praised Mr. Xi, saying he did not blame China for its trade deficit with the United States. Two months later, he launched his trade war with China.

Mr. Trump has “said some nice words,” said Xin Qiang, a U.S.-China expert at Fudan University in Shanghai. “But what’s important is how he translates them into action. His actions are the opposite.”

Other analysts said the apparent lack of a call, so far, could be more easily explained.

Chinese leaders typically do not meet or speak with their foreign counterparts until their subordinates “work out all the thorny issues” for an agreement or a joint statement, said John Gong, a professor of economics at the University of International Business and Economics in Beijing.

Mr. Trump, however, operates more like a business executive, looking to negotiate with world leaders on the fly, as he did during his first presidential term with Kim Jong-un, the North Korean leader.

“Trump is a little bit delusional in the sense that he thinks in his own way, the Western way, that somehow he can do a sales pitch and convince Xi” to do what he wants, Mr. Gong said. “It’s just not going to happen.”

Pope Francis on Tuesday harshly criticized President Trump’s policy of mass deportations and urged Catholics to reject anti-immigrant narratives in an unusually direct attack on the American administration.

In an open letter to American bishops, Francis said that deporting people who often come from difficult situations violates the “dignity of many men and women, and of entire families.”

The pope wrote that he had “followed closely the major crisis that is taking place in the United States with the initiation of a program of mass deportations,” adding that any policy built on force “begins badly and will end badly.”

Francis has long been an advocate for migrants, and has made denouncing their plight a pillar of his papacy. He has called the issue a “shipwreck of civilization” and spoken out repeatedly against what he considers unwelcoming and unchristian migration policies around the world.

Pope Francis had criticized Mr. Trump’s anti-immigration plans when he was a presidential candidate, but the letter was one of the first public and explicit criticisms he has directed at the United States’ president since the election. Experts said it amounted to a steep escalation in the temper of the relationship between the Vatican and the American administration.

“It turns up the heat of the conflict,” said Massimo Faggioli, a professor of theology at Villanova University.

Experts said that by writing an open letter, the pope was also indirectly addressing members of the new American administration, many of whom are Catholic, and specifically, Vice President JD Vance.

Francis appeared to give a riposte to Mr. Vance, who recently talked about the “ordo amoris” — a medieval Catholic theological concept that established a hierarchy of duties that prioritized immediate obligations to one’s family or community over distant needs.

The pope wrote that “Christian love is not a concentric expansion of interests that little by little extend to other persons and groups.” “The true ordo amoris that must be promoted,” he wrote, is “love that builds a fraternity open to all, without exception.”

The pope’s letter, experts said, was also addressed to some bishops and Catholics who have adopted a benevolent stance toward President Trump.

“He wants to avoid that the Church is divided into a Church of the Pope and a Church of Trump,” said Alberto Melloni, a church historian and the director of the John XXIII Foundation for Religious Sciences in Bologna.

Pope Francis has spoken out before against Mr. Trump’s anti-immigration policies.

In 2016, he suggested that Mr. Trump, then a presidential candidate, “is not Christian” because of his campaign promises to deport more immigrants and build a wall along the Mexico border.

Last year, Pope Francis said that both presidential candidates were “against life” — Kamala Harris for her support for abortion rights, and President Trump for closing the door to immigrants. He urged voters to choose the “lesser of two evils.”

But during President Trump’s first term, Francis directed general criticism at the building of walls, but generally refrained from direct attacks on the administration.

This time, Francis has not been shy to criticize President Trump’s policies more directly. On an Italian TV show on the eve of the inauguration, he said that Mr. Trump’s deportation plans, “if true, will be a disgrace.”

During President Trump’s first term, “the Vatican thought Trump was a historical mistake that would be corrected,” Mr. Faggioli said. “Now they know it’s a new era.”

There was no immediate comment from the White House.

In the letter, which was unannounced, Francis urged Catholics to consider human values, not laws or regulations, as the primary compass driving their actions.

“Consider the legitimacy of norms and public policies in the light of the dignity of the person and his or her fundamental rights,” he wrote. “Not vice versa.”

He reminded Catholics that Jesus and his family were migrants to Egypt, and exhorted “all the faithful of the Catholic Church” not to “give in to narratives that discriminate against and cause unnecessary suffering to our migrant and refugee brothers and sisters.”

Other Christian leaders have also criticized President Trump.

During the inaugural prayer service at Washington National Cathedral last month, Bishop Mariann Edgar Budde, the leader of the Episcopal Diocese of Washington, asked President Trump to have mercy on undocumented immigrants, L.G.B.T.Q. children, and others.

The following day, Mr. Trump demanded an apology from the “so-called Bishop” and “Radical Left hard line Trump hater” on his social media platform Truth Social.