High-level cease-fire talks appeared to be gaining momentum on Monday as Arab and American mediators pressed for an agreement to halt the fighting in Gaza and release hostages held by Hamas before President-elect Donald J. Trump assumes office on Jan. 20.

It was still unclear whether the parties had reached a resolution on all the central disputes that have proved insurmountable in previous rounds of negotiations, but officials expressed optimism that a deal was achievable..

On Monday, President Biden suggested an agreement between Israel and Hamas was imminent. “On the war between Israel and Hamas, we’re on the brink of a proposal that I laid out in detail months ago finally coming to fruition,” he said in a foreign policy speech.

Jake Sullivan, the U.S. national security adviser, said there was “a distinct possibility” that Hamas and Israel could agree to a deal this week.

“The question is now can we all collectively seize the moment and make this happen,” he told Bloomberg in an interview.

A Hamas official said in a text message that progress had been made on all issues and that a deal was possible in the coming two days as long as Israel does not change its position at the last minute.

Earlier on Monday, an Arab diplomat said “real progress” was being made in the talks, and two Israeli officials said a draft agreement was awaiting Hamas’s approval, with the next 24 hours seen as being critical.

Other Israeli officials said that the optimal conditions for an agreement had been created, making a breakthrough possible. These officials said that the emerging agreement would allow Israel to maintain a buffer zone in Gaza during its implementation and that Israeli forces would not leave the territory until the release of all hostages.

They also said it would allow displaced Palestinians in southern Gaza to return to the north while unspecified “security arrangements” were enforced.

The Hamas official, the diplomat and the Israeli officials spoke on the condition of anonymity to discuss sensitive diplomacy.

For months, repeated rounds of talks have seen hopes rise only to be dashed days later, with Israel and Hamas each blaming the other for the impasse.

If a deal is achieved, it would bring some respite to Palestinians in Gaza, who have endured miserable conditions in displacement camps and relentless bombardments by Israel, and the families of hostages taken from Israel, who have suffered for months wondering about the fate of their loved ones.

• Hamas leaders want to bring about an end to the Israeli assault, which has severely weakened the group’s armed wing and government, uprooted nearly two million people and reduced cities to rubble. Hamas officials have also said they are seeking a complete Israeli withdrawal from Gaza, the return of displaced people in the south of the enclave to the north, the entry of materials for reconstruction, and freedom for Palestinian prisoners held in Israel. On Monday, Hamas said in a statement that Palestinian prisoners would be freed soon.

• The parties have long been discussing an agreement that would have three stages in what Arab and American officials hope will result in the end of the war. But Israeli officials said on Monday that the deal coming together could have only two phases, with negotiations about the details of the second stage commencing on the 16th day of the first stage.

What are the biggest obstacles?

• A major hurdle to the success of the talks has been the permanency of a cease-fire. While Hamas has demanded a comprehensive end to the war, Mr. Netanyahu has said he wants a “partial” deal that would allow Israel to resume the war after freeing hostages.

• Israel has been demanding vague language in the text of an agreement that leaves room for a resumption of fighting at some point, according to a Palestinian familiar with the matter and two Israeli officials. Mr. Netanyahu has feared that his right-wing coalition partners could take down his government and jeopardize his political future if he agrees to a deal that ends the war, analysts say.

• In a post on X on Monday, Bezalel Smotrich, the Israeli finance minister, described “the emerging deal” as “a catastrophe for the national security” of Israel and declared he would not support it.

• Hamas has not suggested that it would be willing to compromise on its demand to end the war. Osama Hamdan, a senior Hamas official, told a gathering in Algeria last week that there must be “an absolute end to the aggression.”

• Another hurdle has been how far into Gaza Israel will be allowed to carry out military operations in the first phase of an agreement. Israel had wanted the ability to maneuver up to 1.5 kilometers, or about a mile, into the enclave, the two Israeli officials and the Palestinian familiar with the matter said. Hamas had wanted any incursions limited to within 500 meters of the border, according to the Palestinian.

• The Israeli officials, however, have now been saying that the emerging agreement would allow Israel to maintain a buffer zone in Gaza during its implementation and that Israeli forces would not leave the territory until the release of all hostages.

• Israel has demanded a list from Hamas of which hostages are still alive. Without that, Israeli officials say, there can be no agreement on how many Palestinian prisoners Israel would be willing to release in exchange. As of Sunday morning, Israel had not received such a list, according to an official familiar with the matter.

• Last week, Hamas representatives indicated that the group had approved an Israeli list of 34 hostages to be released in the first stage of an agreement, but it did not specify how many of them were alive. On Wednesday, the Israeli authorities announced that the body of one of the hostages whose name appeared on the list— Youssef Ziyadne, 53, an Arab citizen of Israel — had been found in Gaza.

• On Monday, Israeli officials confirmed that the number of hostages to be released in the first stage was 33 and said their assessment was that most of them were alive.

• But Hamas has agreed to Israel’s request to include 11 contested individuals on the list of hostages to be released in the first phase of a deal. Israel classifies them as civilians, but Hamas considers them soldiers, according to the two Israeli officials and the Palestinian. Israel is weighing Hamas’s demand that the 11 be treated as soldiers who would be exchanged for a higher number of Palestinian prisoners than those released for civilian hostages.

Isabel Kershner contributed reporting to this article.