When President-elect Donald J. Trump’s Middle East envoy, Steve Witkoff, met with Prime Minister Benjamin Netanyahu of Israel on Saturday to pressure him on a cease-fire deal in Gaza, there was someone on the speakerphone: Brett H. McGurk, President Biden’s longtime Mideast negotiator.

It was a vivid example of cooperation between two men representing bitter political rivals whose relationship has been best described as poisonous. Rarely if ever have teams of current and new presidents of different parties worked together at such a high-stakes moment, with the fate of American lives and the future of a devastating war hanging in the balance.

“Brett is in the lead,” Mr. Witkoff said last week at Mar-a-Lago, Mr. Trump’s club in Florida, describing the working relationship. That description was accurate by all accounts, even if it did not match what Mr. Trump had said moments before in one of several statements describing his negotiators as critical players.

In fact, Mr. Trump’s threat that “all hell” would break loose if no deal was reached before his inauguration on Monday might have helped motivate Hamas’s leadership to make final decisions. But people familiar with the negotiations said the announcement on Wednesday of a deal to temporarily end hostilities in Gaza was the result of months of work by Mr. McGurk in the Middle East, capped off by several weeks of carefully coordinated efforts by Mr. Witkoff.

Mr. Witkoff, a blunt real estate investor from the Bronx, has largely planted himself in Qatar for the negotiations, knowing that whatever Mr. McGurk negotiated, he would have to execute. In fact, the 33 hostages who will be released under the cease-fire deal may not see freedom until Inauguration Day or after. The cease-fire would expire six weeks later, unless Phase 2 of the agreement kicks in.

By design, the goal was to send a single, unified message that the fighting must end and the hostages held by Hamas must be released. One person familiar with the negotiations, who spoke on the condition of anonymity to describe the discussions, said Mr. McGurk was more involved in hammering out details of the agreement, while Mr. Witkoff’s role was to make clear that Mr. Trump wanted a deal by the time he is inaugurated.

The president-elect has also been setting some early parameters in his dealings with Mr. Netanyahu — who, for all his support of Mr. Trump in the election, was perceived by the Trump camp as dragging his feet on a deal. Mr. Witkoff flew to to Israel from Doha on Saturday — despite the Sabbath — to underscore the message that Mr. Netanyahu had to get on board.

Mr. Witkoff’s work, including the meeting with Mr. Netanyahu, helped Mr. McGurk and the Biden administration to put pressure on both sides during the negotiation, according to the person familiar with the talks.

It was not at all clear that such an arrangement would work in the days immediately after Mr. Trump won a second term.

He and Mr. Biden have barely talked in recent weeks, their already acrimonious relationship weighed down by the Trump team’s determination to clean out the White House career staff and the Biden team issuing last-minute orders to box in the new administration.

The exception has been the Gaza negotiations, according to people on both sides.

One called it a “very close partnership” between Mr. McGurk and Mr. Witkoff and said the result had been an “incredibly effective” process by which the Biden administration finalized a deal that the Trump administration would have to oversee.

That cooperation began soon after Mr. Trump won the election and named Mr. Witkoff to be his envoy to the region. Biden administration officials have said they believe the momentum for a deal began before that, when Mr. Biden helped broker a separate agreement to end fighting between Israel and Hezbollah in Lebanon. That isolated Hamas and helped persuade the group that a cease-fire was in its interests, according to Biden administration officials.

When an Italian journalist was arrested in Iran in December, her boyfriend back home feared she might linger in prison for years. So, amid talk that Iran and Italy were negotiating a prisoner swap that involved the United States, he says, he decided to try to get a message to someone who might be in a position to help.

His name was Elon Musk.

Not only was Mr. Musk close to President-elect Donald J. Trump, a month before the journalist was detained, the tech billionaire had a secret meeting with the Iranian ambassador to the United Nations.

Last week, the journalist, Cecilia Sala, 29, was released from prison in Iran, and days later an Iranian engineer whom Italy had detained on an American extradition request was also freed. The engineer was accused of providing material for drones used in an Iranian-backed militia attack on a U.S. military base that killed three American servicemen.

Mr. Musk helped secure the release of Ms. Sala by reaching out to Iran’s ambassador to the U.N., Amir Saeid Iravani, according to two Iranian officials, one a senior diplomat at the Foreign Ministry, who are both familiar with the terms of the prisoner exchange. They asked that their names not to be published because they were discussing a sensitive issue.

Neither Mr. Musk nor representatives of the Trump transition responded to repeated requests for comment.

How Mr. Musk, an increasingly active if uncredentialed player on the world stage since the Trump victory, came to take up the journalist’s cause remains unclear. He is close with Italy’s prime minister, Giorgia Meloni, who traveled to Mar-a-Lago, the Trump estate in Florida where Mr. Musk has been a regular, and met with the president-elect on Jan 4.

Ms. Meloni said at a news conference last week that Ms. Sala’s release was the result of a “complex work of diplomatic triangulation with Iran, and obviously also with the United States of America.” Her office and the Italian Foreign Ministry declined to comment for this article.

A senior Biden administration official said the American government had not been consulted about the negotiations, had not been given advance word about the releases, and disapproved of the deal. John Kirby, the spokesman for the National Security Council, said that the deal had been “an Italian decision from soup to nuts.”

At the news conference, Ms. Meloni said she did not know what part if any Mr. Musk had played in Ms. Sala’s release. “If he had a role, I am not aware of it,” she said.

The ambiguity highlights the unusual role Mr. Musk has been playing as he sits at Mr. Trump’s side, backing far-right parties in Europe even as he continues to promote his business interests abroad. Italy, for example, is currently exploring a potential deal with Mr. Musk’s Space X to provide secure communications for government and military officials through Starlink. And Ms. Meloni has been one of Mr. Musk’s ever-present European allies, hosting him at her party’s conference in 2023 and attending a gala with him last October.

By the time of Ms. Meloni’s trip to Mar-a-Lago, Ms. Sala’s boyfriend, Daniele Raineri, had already sought out Mr. Musk’s help through an intermediary, he said. In an interview, Mr. Raineri said that he had thought of him because he had read that there was “a channel between Musk and the Iranian diplomats, and that Musk also works in close contact with Trump.”

Mr. Raineri, who is also a journalist, said he fired off a message on Dec. 29 to an Italian computer expert and associate of Mr. Musk’s to ask if he could bring Ms. Sala’s case to the billionaire’s attention and seek his help.

The computer expert, Andrea Stroppa, said in an interview that Mr. Musk had acknowledged the request, but that he did not know whether he had become involved in the case.

In November, weeks before Ms. Sala was arrested, Mr. Musk met for more than an hour with the Iranian ambassador at the Iranian’s residence in Manhattan to discuss defusing tensions between Tehran and Washington as a new administration prepared to take power.

The Iranian officials interviewed for this article said that Mr. Musk contacted the ambassador again shortly after Prime Minister Meloni visited Mar-a-Lago.

The prisoner exchange between Iran and Italy then unfolded rapidly. Iran released Ms. Sala on Jan. 8, and four days later, Italy freed the Iranian engineer, Mohammad Abedini Najafabadi.

Mr. Abedini had been detained at the request of the U.S. Justice Department. A federal court in Massachusetts accused him of procuring drone technology for Iran that was used in the attack on the American base, which was in Jordan, in January 2024. The two Iranian officials said that when Mr. Musk spoke with the ambassador, he requested that Iran release Ms. Sala and reassured him that the United States would not pressure Italy to extradite the Iranian engineer.

Iran’s mission to the U.N. declined to comment on the recent engagement between Mr. Musk and the ambassador. In a statement, it said the two detainees had been released as a result of “bilateral cooperation and coordinated efforts of the political and intelligence sectors of Iran and Italy.”

All sides are being circumspect in public about what transpired, but to many observers the speedy release of the prisoners following the meeting between Mr. Trump and Ms. Meloni suggested that the topic had been discussed and a resolution was reached.

“The most likely reconstruction is that she got a signal of understanding from Trump that the incoming administration would not raise huge problems if it released Abedini,” said Ferdinando Nelli Feroci, a former Italian diplomat.

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Italy detained Mr. Abedini in mid-December as he was transiting through Milan’s airport. Three days later, agents from the intelligence wing of the Revolutionary Guard raided Ms. Sala’s hotel room in Tehran and threw her into solitary confinement at Evin prison. She had traveled to Iran on a journalist visa.

A member of Iran’s Revolutionary Guards, and the two Iranian officials said the journalist had been arrested to pressure Italy to release Mr. Abedini. Iran has made the detention of foreign and dual nationals a centerpiece of its foreign policy for nearly five decades.

“It doesn’t matter who you are,” Shahin Modarres, a Rome-based expert on Iran and international security, said in a telephone interview. “You can be a journalist, a diplomat, a tourist.” He said, “What matters is if Iran thinks it can use you as leverage.”

Ms. Sala said in an interview that she had been held in a cell with no mattress and had slept on the floor with one blanket on top and another below. For weeks, she was denied her glasses, and throughout her detention she did not see a human face. She could hear the sound of other inmates crying and vomiting, she said, and was blindfolded and interrogated for hours nearly every day, she said.

What she feared the most, she said, was “that I would go insane.”

Mr. Raineri, her boyfriend, said that on Jan. 2, Ms. Meloni told Ms. Sala’s mother that within the next 48 hours there would be an important development.

Mr. Stroppa, the programmer, dropped hints on his account on X. He posted a portrayal of Mr. Trump, Ms. Meloni and Mr. Musk in ancient Roman attire on the day the prime minister visited Mar-a-Lago. And on the day Ms. Sala was released, he posted an AI-generated photo of Mr. Musk eating spaghetti with an Italian flag emoji.

Ms. Sala said she understood that Italy’s government did what it had to do to free her. “And whoever it was they needed to talk to, they talked to,” she said.

Ryan Mac and Mattathias Schwartz contributed reporting.