The two officials said the response from Hamas was a serious one, and that it was now up to Israel to decide whether to enter into an agreement. The proposal, they said, calls for Hamas to free hostages — women, the elderly and those in need of medical treatment — in return for a 42-day cease-fire and the release of a much larger number of Palestinian prisoners. Israel had sought 33 hostages, but it is not clear how many women and elderly are still alive, and the first tranche could end up including remains.

That would be the first of three phases of reciprocal actions from each side. In the second phase, the two sides would work toward reaching a “sustainable calm,” which would involve the release of more hostages, the officials said. Both officials acknowledged that the warring parties would likely clash over the definition of “sustainable calm.”

One of the officials, in the Middle East, said that Hamas viewed the term as an end to the war, with Israel halting its military actions and withdrawing troops from Gaza. The officials said that Israel’s prime minister, Benjamin Netanyahu, was expected to push back against that definition.

One official said that the negotiating parties agreed to the term “sustainable calm” weeks ago, after Israel objected to any reference to a “permanent cease-fire.” Israeli officials have consistently said they oppose any agreement that explicitly calls for that or for an end to the war.

Mr. Burns has been the main representative for the United States in the negotiations, and he is in the region to work on the proposals and counterproposals. Qatari and Egyptian mediators spoke with him on Monday about the changes that Hamas was ready to accept, the two officials said. Hamas said that Arab mediators had put forward the changes, but one official said that Hamas had suggested them. Mr. Burns is expected to attend the talks in Cairo on Tuesday.

The Israeli prime minister’s office said that while the new proposal failed to meet Israel’s demands, the country would still send a working-level delegation to talks in hopes of reaching an acceptable deal. A U.S. official said the purpose of the talks in Cairo was to negotiate the amendments proposed by Hamas and talk through remaining issues.

Qatar’s Foreign Ministry said that a Qatari delegation would also attend the talks on Tuesday, and expressed “hope that the talks will culminate in reaching an agreement for an immediate and permanent” cease-fire, an exchange of hostages and prisoners, and a “sustainable” flow of aid into all of Gaza.

Israel announced on Monday that its war cabinet had voted unanimously to continue with its military action in Rafah in order to exert pressure on Hamas. That announcement and the start of any offensive in the city could jeopardize the prospects for an agreement. Mr. Netanyahu said last week that he would carry out an offensive in Rafah “with or without” an agreement.

U.S. officials say they oppose any such operation without a proper plan from Israel to mitigate civilian casualties and a humanitarian crisis. One U.S. official said the strikes that the Israeli military carried out in eastern Rafah on Monday appeared to be part of a smaller operation, and not necessarily the opening moves of a larger assault. More than one million Palestinians have sought shelter in Rafah as they fled other parts of Gaza under attack by Israel.