U.S. officials made a new offer to the Taliban over the weekend to try to secure the release of Americans held in Afghanistan, according to people familiar with the matter.

American officials declined to discuss the negotiations or the offer. But people briefed on the conversations said the White House had been working on a deal to get the Americans back in exchange for Muhammad Rahim, an Afghan who has been held at Guantánamo Bay since 2008.

The U.S. government has said Mr. Rahim was a senior aide in Al Qaeda, but others have cast doubt on his role in the organization, suggesting that he was a courier and a translator and would not pose a threat to the United States if he were released.

The Taliban have acknowledged holding two Americans. One of them, George Glezmann, a former airline mechanic, was touring Afghanistan when he was taken prisoner in December 2022. The other, Ryan Corbett, who had long lived in Afghanistan with his family before the fall of the U.S.-backed government, was on a business trip when he was seized in August 2022.

The Taliban have refused to say whether they are holding a third American whose return is also sought by the U.S. government. That person, Mahmood Habibi, a naturalized American, was taken captive soon after the U.S. strike in Afghanistan in 2022 that killed Ayman al-Zawahri, the leader of Al Qaeda.

Mr. Habibi’s family has said that he was arrested with 30 other people who all worked for the same American firm, Asia Consultancy Group, on suspicion that the company was involved in the U.S. drone strike that killed the Qaeda leader. U.S. officials would not discuss whether Mr. Habibi had a role in the strike, nor is his condition known.

But his father, Ahmadullah, and brother, Ahmad, have denied that he was involved. Mr. Habibi was out of the country at the time of the attack and arrived in Afghanistan days later, they said. They have also said, based on “several independent sources in Afghanistan,” that he was alive and last known to be in custody of the government’s General Directorate of Intelligence.

Sean Savett, a National Security Council spokesman, said the Biden administration had negotiated the return of more than 75 Americans held overseas. He said officials were “working around the clock” to ensure the return of Mr. Glezmann, Mr. Corbett and Mr. Habibi.

“The administration will do so throughout the remainder of the term,” he added.

The administration made an earlier offer to the Taliban to secure the release of the three men on Nov. 14, a proposal that The Wall Street Journal reported earlier.

Mr. Rahim has never been charged with crimes while held in Guantánamo. James G. Connell III, a defense lawyer who has represented Mr. Rahim for his detention review hearings, said that Mr. Rahim was willing to be traded but that the U.S. government had not reached out to them.

“I have never received official confirmation from any government that talks are ongoing, or were ongoing, for that matter,” Mr. Connell said.

Mr. Rahim is not currently eligible for transfer out of Guantánamo. A federal national security review panel has repeatedly deemed him too dangerous to be released, most recently on Nov. 21, 2023.

But U.S. officials have discussed transferring him to Qatar as part of a prisoner swap, according to the people briefed on the discussions.

The Defense Department is legally required to notify Congress 30 days before a detainee is released from Guantánamo.

But the Obama administration ignored that requirement when it sent five Taliban prisoners from Guantánamo to Qatar in return for the release of Sgt. Bowe Bergdahl in 2014. Like Mr. Rahim, none of those prisoners were approved for release through the national security review panel process.

Republicans criticized the Bergdahl trade, and a Government Accountability Office study concluded that the Obama administration broke the law.

But Dennis M. Fitzpatrick, a lawyer for the Glezmann family and a former terrorism prosecutor, said that Mr. Rahim was improperly classified as ineligible to be released and that the government had long exaggerated the danger he posed.

“He is no longer a national security threat,” Mr. Fitzpatrick said. “He was never operational with respect to being a fighter or organizer. He’s also not an intelligence risk to the United States, nor does he have any intelligence value. If he’s not operational and there’s no intelligence issue, then he’s not a national security threat.”

In a letter sent to the White House last July, Aleksandra Glezmann, Mr. Glezmann’s wife, spoke about President Biden’s long commitment to public service while she pushed for a deal.

“I beseech you to intervene personally in George’s matter and do everything in your power to bring him home,” she wrote. “We understand the Taliban poses extreme geopolitical challenges for the United States, and we do not suggest that securing George’s release is easy. But you did not enter a life of public service to do easy things.”

Mr. Corbett and his family had lived in Kabul between 2010 and 2021. After leaving Afghanistan during the American withdrawal, Mr. Corbett returned for a trip, to support his Afghan employees who were continuing his microloan and consulting business.

Since his capture, his wife, Anna Corbett, and their children have been lobbying the White House and officials in Washington to press for his release.

Both families met with Jake Sullivan, the national security adviser, late last year.

Over the last month, President-elect Donald J. Trump has repeatedly discussed his desire for the United States to purchase Greenland, and during a long news conference on Tuesday he refused to rule out using military force to accomplish that goal.

In a far more subtle act of geopolitics that came ahead of Mr. Trump’s remarks, Denmark’s newly proclaimed king, Frederik X, implemented changes to the Danish royal coat of arms that reaffirm his kingdom’s commitment to Greenland, a sovereign territory.

The Danish royal coat of arms — a more elaborate symbol than the kingdom’s national coat of arms — had previously featured a panel with three crowns that represent the Kalmar Union between Denmark, Norway and Sweden. Those crowns have now been removed, with separate panels being dedicated to Greenland (represented by a polar bear) and the Faroe Islands (represented by a ram). In the previous royal coat of arms, Greenland’s polar bear was in the same lower left panel, but was far smaller.

A statement from The Royal House of Denmark said of the new design that “the Faroe Islands and Greenland have each gotten their own field, which strengthens the Realm’s prominence in the royal coat of arms.” The royal symbol is a coat of arms for the king’s personal use, and also a state symbol.

According to the statement, the royal coat of arms received a public showing last week when it was featured on a flag raised at Amalienborg Palace in Copenhagen for the royal family’s New Year’s banquet, where it also appeared on royal vehicles as guests arrived.

Lars Hovbakke Sorensen, an expert on the Danish royal family, said in an interview with TV 2 that the design alteration’s understated nature didn’t make it less of a statement.

“It is important to signal from the Danish side that Greenland and the Faroe Islands are part of the Danish realm, and that this is not up for discussion,” he said. “This is how you mark it.”

If tweaking a heraldic design seems like a restrained way to convey a political message, Mr. Trump in comparison has been anything but subdued in stressing his interest in American territorial expansion into Greenland.

In a Truth Social post last month, in which he announced that he was appointing Ken Howery, the diplomat and a co-founder of PayPal, as his choice for ambassador to Denmark, Mr. Trump said: “For purposes of National Security and Freedom throughout the World, the United States of America feels that the ownership and control of Greenland is an absolute necessity.”

He stressed his intent again in a post on Monday, in which he said that his son, Donald Trump Jr., would soon be traveling to Greenland. “Greenland is an incredible place,” he said, “and the people will benefit tremendously if, and when, it becomes part of our Nation.”

When the younger Mr. Trump arrived in the island’s capital of Nuuk on Tuesday, along with an entourage that included the conservative activist Charlie Kirk, he told reporters awaiting him at the airport that he was making a brief visit as a tourist.

But his visit was seen by some in the region as a provocation. Spurred to comment more directly, perhaps in a way that redesigning a royal coat of arms can’t, Denmark’s prime minister, Mette Frederiksen, addressed the public in a television appearance.

“Greenland belongs to the people of Greenland,” she said. “Our future and fight for independence is our business.”

Crossings into the United States from Mexico dropped sharply last year. But countries south of the U.S. border are waiting nervously to see if President-elect Donald J. Trump orders mass deportations.

The possibility that millions of undocumented immigrants could be expelled — what would be the largest deportation program in American history — has sent shock waves through Latin America and sowed confusion among migrants and asylum seekers.

“We see dark times coming for the migrant community,” said Irineo Mujica, the Mexico director of People Without Borders, a transnational advocacy group. “Anyone who falls prey to the Trump administration is now going to be devoured, chewed up and spat out.”

What is the situation at the U.S.-Mexico border?

Mr. Trump has said that Mexico is allowing an “invasion” of migrants into the United States. But the current situation on the ground tells a different story.

Unlawful crossings at the U.S.-Mexico border have been declining since June, when President Biden issued an executive order to essentially block undocumented migrants from receiving asylum at the border.

That month, U.S. Border Patrol officials recorded 130,415 apprehensions of migrants — a sharp drop from the more than 170,710 recorded the previous month. The numbers in November were even lower: U.S. officials recorded 94,190 people.

That is a stark shift from a year ago. Illegal crossings for November 2023 rose above 242,300, a record at the time.

How have the U.S. and Mexico reduced crossings?

Critics who argue that asylum is legal and a basic human right say Mr. Biden’s move was a short-term fix for a complex issue.

As part of Mr. Biden’s order, restrictions are to be lifted when the number of people trying to cross illegally each day drops below 1,500 for one week. That has not happened. But it has sharply brought down border crossings and allowed officials to deport those who cannot prove they would be endangered if they returned to their countries.

Mexico has also clamped down on people heading to the U.S. border.

It has deployed National Guard troops to immigration checkpoints from north to south. More recently, the authorities have bused migrants farther south into Mexico — in what officials and scholars call a migratory merry-go-round. They have prevented them from hopping onto trains heading north and have broken up caravans, which no longer reach the U.S. border.

In 2023, Mexico paused the issuance of humanitarian cards that allowed asylum seekers to study, work and get access to basic services in Mexico. Under the law, they are supposed to stay in the state where they apply for asylum. But many use the cards to move north without being detained, officials say.

As a result of the stoppage, between Oct. 1 and Dec. 26, 2024, Mexican security forces said, they detained over 475,000 migrants, nearly 68 percent more than the number apprehended during the same period in 2023, government data show.

What is the status of migrants waiting in Mexico?

As Mexico’s strategy has shifted, many migrants have become stranded.

“By not giving them cards, they could no longer access public services or enter the legal market,” said Andrés Ramírez Silva, who until September was the head of the country’s Commission for Refugee Assistance.

The situation is unsustainable, advocacy groups warn. More migrants have become easy prey for organized crime groups, which extort them.

“Many people keep arriving” in Mexico, said Mauro Pérez Bravo, the former president of the citizen council of the National Migration Institute. But they live in “vulnerable conditions,” he added, working low-paid jobs or sleeping in shelters, junkyards, construction sites or on the street.

How is Mexico preparing for mass deportations?

Mexican border states have been working in coordination with the federal government to set up shelters to provide food and health services.

They have been making transportation arrangements for Mexicans who wish to return to their home states. In Tijuana, a border town south of San Diego, city officials have been coordinating with churches, bus companies and humanitarian groups to prepare for arrivals, said José Luis Pérez Canchola, director of the city’s migration services office.

He worries that mass deportations from the United States could further strain Tijuana’s resources for migrants, noting that many are likely to be unaccompanied minors or in need of medical attention.

Making sure people do not stay long in Mexican border cities like Ciudad Juárez is a major priority, said María Eugenia Campos, governor of Chihuahua state, which shares an extensive border with Texas and New Mexico.

“The state of Chihuahua cannot become a sanctuary state” for migrants and deportees, she said.

Until this month, Mexico’s president, Claudia Sheinbaum, had said the country would not accept foreign deportees. On Friday, she signaled otherwise.

“We are going to ask the United States that, as far as possible, the migrants who are not from Mexico can be sent to their countries of origin — and if not, we can collaborate through different mechanisms,” she told reporters, adding that her government had “a plan,” without offering details.

Have the factors driving migration changed?

Not really.

About 392,000 Mexicans were displaced as a result of conflict and violence in 2023, according to the Internal Displacement Monitoring Center, which compiles data from federal governments. That was the highest figure since record-keeping began in 2009.

The situation is somewhat similar in Central America. In some countries, criminal gangs and drug cartels have led many to flee.

Honduras had more than 240,000 people internally displaced because of insecurity by the end of 2022, according to a recent report by the International Organization for Migration.

In Guatemala, factors that drive people out — inequality, poverty, climate change, economic instability and violence — have not improved much despite the election of a new president, Bernardo Arévalo, an anticorruption crusader, said Aracely Martínez, a migration researcher at the Universidad del Valle in Guatemala City.

“We have a new government whose campaign proposed fundamental changes, but we still do not see direct results,” she said.

Still, the number of Guatemalans recorded at the U.S.-Mexico border decreased to nearly 8,000 in November from more than 20,000 in January 2024, when Mr. Arévalo took office, U.S. Border Patrol data indicate.

What is the situation elsewhere?

Venezuela and Cuba, which have faced harsh U.S. sanctions, are likely to refuse large numbers of deportation flights.

Honduras, Guatemala and El Salvador signed asylum agreements with the first Trump administration to require people, mostly asylum seekers from Latin America, to first take refuge in those three countries before applying in the United States, though the policy was not put in place in Honduras and El Salvador.

The most concrete pushback against Mr. Trump’s vow of mass deportations has come from President Xiomara Castro of Honduras, who said this month that bases housing U.S. military forces “would lose all reason to exist” in her country if he carried out his promise.

In Guatemala, the government denied as “fake” reports that officials were open to receiving deported foreigners.

Panama in December reported 4,849 people migrating through the perilous Darién Gap — the stretch of jungle that has become a popular migrant route — the fewest numbers in more than two years. Some experts see that as a likely sign of migrants delaying their plans until after Mr. Trump’s election, as well as Panama’s efforts to limit undocumented migration taking effect.

“We can’t claim victory, but for the moment we are curbing — the figures say so — the flow of migrants,” Javier Martínez Acha, Panama’s foreign affairs minister, said in an interview.

In El Salvador, Mr. Trump may find an ally in President Nayib Bukele, who is close to members of the president-elect’s inner circle.

The Bukele administration has not spoken publicly about mass deportations. Asked about specific preparations for mass deportations, an operator with one of the call centers El Salvador set up to provide information to Salvadorans in the United States said, “We can’t get ahead of ourselves.”

Jody García contributed reporting from Guatemala City, Gabriel Labrador from San Salvador and Mary Triny Zea from Panama City.