The director of the C.I.A. is set to make a renewed push for negotiations over the release of hostages held in Gaza and an extended cease-fire, according to U.S. officials, who said he would meet with senior Israeli, Egyptian and Qatari officials in Europe.

The officials hope to close a wide gap between Israel and Hamas, especially on two issues: the length of any pause in fighting and the fate of Hamas leaders in Gaza, according to officials briefed on the talks.

Here is a look at where the talks stand.

How are the negotiations going?

A weeklong truce in November allowed the release of more than 100 of the hostages abducted in Hamas’s Oct. 7 attack on Israel; 240 Palestinian prisoners were released as part of that deal. Since then, both sides have staked out seemingly intractable positions for another such agreement.

The talks have advanced in fits and starts, with the leader of Israel’s Mossad intelligence agency meeting with Qatari officials in both Qatar and Europe. Many of Hamas’s political leaders are based in Qatar. Egypt, which borders the Gaza Strip, has also played a key role.

William J. Burns, the C.I.A. director, is going to meet with senior Israeli, Egyptian and Qatari officials, according to U.S. officials. They said Israel’s apparent willingness to agree to a longer cessation of hostilities as part of any further hostage releases has created a new opening for negotiations.

Any new deal would likely include phased releases of hostages, though the White House is hoping that a bigger deal, leading to the release of the remaining hostages, might be possible. The officials spoke on condition of anonymity to discuss the sensitive diplomacy.

In recent weeks, the mediators have put forward multiple plans, though with little evident progress. Durations for a proposed cease-fire have ranged from weeks to months. Leaks of some of the proposals to the press have prompted controversy in Israel, where right-wing politicians said they would oppose plans they said would end the war prematurely.

Brett McGurk, the top Middle East coordinator at the White House, headed back to the region on Sunday to work on freeing hostages, according to two American officials who spoke on the condition of anonymity.

What terms are being floated for a new cease-fire?

Hamas officials say they will only release the remaining hostages in Gaza, believed to number more than 100, as part of a comprehensive cease-fire. Benjamin Netanyahu, Israel’s prime minister, said on Sunday that he would not accept any deal for a permanent cease-fire that left Hamas in control of Gaza.

Under one recent framework for a deal, mediators proposed a phased release of the remaining hostages and Palestinian prisoners, with the goal of achieving a stable cease-fire, a senior Western diplomat and a regional diplomat said.

What are the sticking points?

The biggest stumbling block is whether a cease-fire would be designated as temporary, like the last one, or permanent.

Israeli officials have suggested they might consider a permanent cease-fire if Hamas’s Gaza leadership leave the strip and go into exile, the two diplomats said.

Hamas officials have rejected that idea. “Hamas and its leaders are on their land in Gaza,” Husam Badran, a senior Hamas official, said in a text message. “We won’t leave.”

Another possible impediment to this plan: Mr. Netanyahu said in November that he had told Mossad “to act against the heads of Hamas wherever they are,” likely raising fears within Hamas about whether the leaders would be less safe outside Gaza.

What happens after the war ends?

Another negotiating track involves the future of Gaza after the guns fall silent.

Biden administration officials have said they hope the Palestinian Authority, which administers parts of the Israeli-occupied West Bank, will return to control Gaza. U.S. officials would like to see both areas included in a future Palestinian state.

Hamas seized control of Gaza in 2007, expelling the rival Fatah party, which dominates the Palestinian Authority. If Hamas remains in Gaza after the war, it would likely prove a formidable obstacle.

Mr. Netanyahu has largely ruled out the return of the Palestinian Authority, in its present form, to ruling Gaza. He has also indicated that he would oppose the establishment of an independent Palestinian state after the war.

Complicating matters further, the Palestinian Authority, led by Mahmoud Abbas, faces severe internal challenges. Polls regularly show that most Palestinians want Mr. Abbas to resign. He was last elected to a four-year term in 2005, and critics accuse him of presiding over an increasingly autocratic government that has failed to end Israeli rule.

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