JERUSALEM, Israel — In a sign that mediators believe a Gaza cease-fire deal is imminent, an American official said Friday that Mideast negotiators are working out logistics for the potential release of hostages and distribution of aid as part of any agreement to end the Israel-Hamas war.

The official, who spoke to reporters on condition of anonymity in keeping with rules set by the White House, said the proposal currently on the table basically bridges every gap between Israel and Hamas and mediators were making preparations before a final deal is approved.

It was unclear what measures were being taken, but the official said a new “implementation cell” was being established at Cairo in advance. The cell would focus on logistics, including freeing hostages, providing humanitarian aid for Gaza and ensuring that the terms of the pact are met, the official said.

The comments came hours after mediators expressed hope that a deal was within reach. They said that two days of talks had wrapped up in Qatar and that they plan to reconvene in Cairo next week to seal an agreement to stop the fighting.

Israel issued a vague statement saying it appreciated the mediators’ efforts, and a statement from Hamas did not sound enthusiastic about the latest proposal to end the devastating 10-month war and free Israeli hostages held in Gaza. A cease-fire is seen as the best hope for heading off an even larger regional conflict.

President Biden seemed optimistic, saying, “We are closer than we’ve ever been” to an agreement. Mr. Biden has expressed optimism for a deal before, only for talks to break down.

“As of an hour ago, it’s still in play,” he said, as he was traveling to spend the weekend at the Camp David presidential retreat. “It’s far from over. Just a couple more issues, I think we got a shot.”

Both sides agreed in principle to the plan Mr. Biden announced on May 31. Hamas, though, has proposed amendments, and Israel has suggested clarifications, leading each side to accuse the other of trying to tank a deal.

The American official said the latest proposal is the same as Mr. Biden’s with some clarifications based on ongoing talks. The way it’s structured poses no risk to Israel’s security but enhances it, the official maintained.

Hamas has rejected Israel’s demands, which include a lasting military presence along the border with Egypt and a line bisecting Gaza where it would search Palestinians returning to their homes to root out militants.

Hamas quickly cast doubt on whether an agreement was near.

In a statement, the militant group said the latest proposal diverged significantly from the previous iteration they had agreed to in principle, implying they were not disposed to accept it.

The Israeli prime minister’s office issued a statement saying it “appreciates the efforts of the U.S. and the mediators to dissuade Hamas from its refusal to a hostage release deal.”

Secretary Blinken planned to travel to Israel over the weekend to “continue intensive diplomatic efforts” toward a cease-fire and to underscore the need for all parties in the region to avoid escalation, State Department spokesman Vedant Patel said. Mr. Blinken was expected to meet with Prime Minister Netanyahu on Monday to discuss the new deal, said an Israeli official.

The new push for an end to the Israel-Hamas war came as the Palestinian death toll in Gaza climbed past 40,000, according to Gaza health authorities, whose counts do not distinguish between civilians and combatants. Fears were still high that Iran and Hezbollah militants in Lebanon would attack Israel in retaliation for the killings of top militant leaders.

International mediators believe the best hope for calming tensions would be a deal between Israel and Hamas to halt the fighting and secure the release of Israeli hostages. International diplomacy to prevent the war from spreading intensified Friday, with the British and French foreign ministers making a joint trip to Israel.

Israel’s foreign minister, Israel Katz, said in a statement that he told his British and French counterparts that if Iran attacks Israel, Israel expects its allies not just to help it defend itself, but to join in attacking Iran.

He also warned Iran — which backs Hamas, Hezbollah and Houthi rebels in Yemen, all of whom have attacked Israel since the Gaza war started — to stop the attacks. “Iran is the head of the axis of evil, and the free world must stop it now before it’s too late,” Mr. Katz said on X.

The war began when Hamas-led terrorists stormed across the heavily guarded border on October 7, killing around 1,200 people, mostly civilians, and abducting 250 to Gaza. More than 100 were released during a weeklong cease-fire in November, and around 110 are believed to still be inside Gaza, though Israeli authorities believe around a third of them are dead.

Israel’s military spokesman, Rear Admiral Daniel Hagari, said Thursday that Israel had killed more than 17,000 Hamas militants in Gaza in the war. Diplomats hoped a cease-fire deal would persuade Iran and Lebanon’s Hezbollah to hold off on retaliating for the killing of a top Hezbollah commander in an Israeli airstrike in Beirut and of Hamas’ top political leader in an explosion in Tehran that was widely blamed on Israel.

The mediators have spent months trying to hammer out a three-phase plan in which Hamas would release the hostages in exchange for a lasting cease-fire, the withdrawal of Israeli forces from Gaza and the release of Palestinians imprisoned by Israel.

While talks were ongoing, Israel continued its offensive in Gaza. On Friday it dropped leaflets asking civilians to evacuate from areas in northern Khan Younis and eastern Deir al-Balah, saying forces plan to respond to rocket fire that targeted Israel. After the orders were given, airstrikes hit some areas of Khan Younis, sending people fleeing. A video showed plumes of black smoke rising into the air after loud booms.

Also Friday, President El-Sisi of Egypt spoke to Mr. Biden and agreed to intensify joint efforts in the coming days to reach an agreement, said a spokesman for the presidency. Mr. El-Sisi also urged regional self-restraint.

In a clear message to Israel, Hezbollah released a video, with Hebrew and English subtitles, showing underground tunnels where trucks were transporting long-range missiles.

A Hezbollah official, who spoke on condition of anonymity because he was speaking about military affairs, said the missiles in the video have a range of about 86 miles, capable of reaching deep inside Israel. Hezbollah has tens of thousands of rockets, missiles, and drones that the group says give it the ability to hit anywhere in Israel. Hezbollah started attacking Israel on October 8 and says it will stop only when the Gaza war ends.