JERUSALEM — The United States and its allies said Friday they have presented a “bridging” proposal to Hamas and Israel for a cease-fire and hostage deal, potentially bringing within reach by the end of next week an agreement that officials hope can put an end to 10 months of fighting in the shattered Gaza Strip.

The United States, Qatar and Egypt, in a joint statement after two days of talks in the Qatari capital, Doha, expressed hope that months of fitful negotiations would finally yield an agreement to end a conflict that has killed tens of thousands and plunged the Middle East into a renewed cycle of violence and instability.

“We’re much, much closer than we were three days ago,” President Biden told reporters in the Oval Office. “I don’t want to jinx anything,” he added, but said, “We’re close.”

The joint statement from the mediating nations said their proposal “bridges remaining gaps” between Israel and Hamas and would allow “a swift implementation of the deal.”

But the fate of the Biden administration’s latest effort to end the war – and, officials hope, avert an Iranian attack on Israel over a spate of assassinations on militant leaders – remained uncertain, as Hamas accused Israel of seeking to prolong the war and mediators stopped short of affirming that ongoing points of disagreement had been resolved.

The past two days of talks in Doha – which U.S. officials said were more productive than months of previous discussions – focused largely on details of the exchange of hostages in Gaza for Palestinian prisoners in Israel. In coming days, lower-level officials will seek to hammer out those particulars and other issues and will work to establish an “implementation cell” to oversee execution of a possible agreement, including provisions to alleviate dire humanitarian conditions in Gaza and monitor compliance of parts of the deal.

Before the end of next week, senior officials will reconvene in Cairo “with the aim to conclude the deal under the terms put forward today,” the mediators’ statement said. Biden also spoke Friday with the leaders of Qatar and Egypt.

Secretary of State Antony Blinken will travel to Israel on Sunday, seeking to usher things toward a conclusion.

The on-again, off-again talks are “now in the end game,” a senior U.S. official told reporters after the talks. “There’s really a new spirit here to drive this to a conclusion.” The official, like others in this article, spoke on the condition of anonymity because of the sensitivity of the negotiations.

The office of Israeli Prime Minister Benjamin Netanyahu referred to an earlier framework that had been embraced by the Biden administration in May. “Israel’s fundamental principles are well known to the mediators and the U.S.,” the office said. “Israel hopes that their pressure will lead Hamas to accept the principles of May 27, so that the details of the agreement can be implemented.”

Senior Hamas officials, who did not take direct part in the talks in Doha, said the outline they were briefed on did not align with frameworks to which they had agreed. Ghazi Hamad, a member of Hamas’s political bureau, said the Israeli leader had introduced new conditions and was “practicing deception.”

“Netanyahu sets new conditions and undermines what was previously agreed upon,” he told al Mayadeen, an Arab television channel. “His goal is to continue the war.”

But Washington wasn’t alone in expressing optimism about the latest round of negotiations, with some Arab officials crediting Biden’s sense of urgency with moving things along.

“I am more hopeful compared to any other time before,” said a former senior Egyptian official with knowledge of the negotiations. “They are the ones who most want to get it over and done with more than anyone else.”

After the first day of talks Thursday, Qatar’s prime minister assessed that progress was being made and implored Iran’s leaders to hold off on an expected assault on Israel, to avoid scuttling an emerging deal, according to two diplomats familiar with the conversation.

The phone call from Mohammed bin Abdulrahman Al Thani to Iran underscored the stakes of the negotiations, which included Israel.

Mohammed “encouraged Iran to de-escalate and stressed the need for calm” during the call, one of the diplomats said. The second diplomat said the Qatari prime minister warned Iran to consider the grave consequences of waging an assault at the very moment there are signs of diplomatic progress.

An Israeli official also said progress had been made in the first day of talks, justifying Netanyahu’s decision to keep his negotiators in Doha overnight.

The official and the two diplomats declined to specify where Israeli officials may have adjusted their position on the proposed deal, which includes a six-week cease-fire, the release of scores of hostages and the withdrawal of Israeli forces from major population centers in Gaza. Israel previously insisted on additions to a framework announced by Biden in May.

For weeks, Iran and the Lebanese militant group Hezbollah have threatened to retaliate against Israel for the assassinations of Hamas political leader Ismail Haniyeh in Tehran and Hezbollah senior commander Fuad Shukr in Lebanon. Israel has neither confirmed nor denied its role in Haniyeh’s killing – though it privately told U.S. officials that it was responsible.

The Israel Defense Forces has been on high alert, and the White House warned Thursday that a strike could happen imminently and without notice.

“We have information … that an attack could come with little or no warning, and certainly could come in coming days, and we have to be ready for it,” National Security Council spokesman John Kirby said in a phone briefing.

In the Doha talks, the United States was represented by CIA Director William J. Burns and White House Middle East coordinator Brett McGurk. Egypt was represented by its intelligence chief, Abbas Kamel, and Israel by Mossad head David Barnea as well as Shin Bet chief Ronen Bar and Maj. Gen. Nitzan Alon. Mohammed was in attendance for Qatar.

Husam Badran, a member of Hamas’s political bureau, said Thursday on social media that “any agreement must achieve a comprehensive cease-fire, a complete withdrawal from Gaza, the return of the displaced and the reconstruction, in addition to a prisoner exchange deal.”

During previous rounds of talks, Israel put forward additions to the framework announced by Biden, including its military retaining indefinite control of the Philadelphi Corridor, a buffer zone between Gaza and Egypt, according to officials familiar with the process. It also insisted on setting up checkpoints to process Palestinians returning to their homes in northern Gaza. Hamas had appeared to meet the conditions in the proposal announced by Biden, U.S. officials said.

An Israeli official said Thursday that Netanyahu is adamant that the IDF remain in the Philadelphi Corridor.

Israeli media, which first reported on the Qatari prime minister’s conversation with Iran’s leaders, said a breakthrough in the talks is related to the Netzarim Corridor, a four-mile road just south of Gaza City that runs east to west, from the Israeli border to the Mediterranean Sea. Hamas has made Israel’s withdrawal from the area a central demand in cease-fire negotiations.

The U.S. official did not specify whether mediators’ latest proposal includes those demands. He acknowledged that any agreement would not fully meet the aspirations of either side.

“If you continue to negotiate for months and months and try to get a perfect deal for every last drop of blood from the stone, you risk having no hostages,” he said. “That is not acceptable to us. I think it’s not acceptable to anybody.”