When the cease-fire agreement between Israel and Hamas was announced in January, Israelis and Palestinians burst into simultaneous celebrations, optimistic after 15 months of war.

Now, with the first phase of the deal over on Sunday and Israel introducing an entirely new proposal that Hamas has already rejected, concern is rising that the fighting that reduced Gaza to rubble, killed tens of thousands of Palestinians and threatened the lives of hostages could resume.

As the cease-fire teeters, both Hamas and Israel are pursuing two paths, one diplomatic and another military.

On the diplomatic front, Hamas is insisting on the implementation of the second phase of the original agreement, which calls for an end to the war, a full Israeli withdrawal from Gaza and the release of more hostages and prisoners.

Israel, though, has made a new proposal for a seven-week extension of the current cease-fire, during which Hamas would be required to release half the remaining living hostages as well as the remains of half the deceased ones. Prime Minister Benjamin Netanyahu of Israel on Sunday attributed the proposal to the work of President Trump’s Middle East envoy, Steve Witkoff.

For weeks, Israel has been sending signals that it wasn’t interested in moving forward with the second phase of the agreement. While the two sides agreed to the second phase in principle, they never worked out the details and have staked out irreconcilable visions.

Mr. Netanyahu has said repeatedly that Hamas’s government and military wing must be dismantled, a position shared by his right-wing coalition partners in the government. Hamas has suggested it was willing to give up civilian governance of Gaza but has firmly rejected dissolving its military wing, a critical source of its power in the enclave.

The new proposal, as described on Sunday by Mr. Netanyahu, appears to be an attempt to replace the cease-fire deal with terms that would enable Israel to bring home dozens of hostages and remains of hostages without committing to the end of the war.

But the suggestion, analysts said, may be an effort to shake up the cease-fire talks in a way that breaks the deadlock between Israel and Hamas, at least temporarily.

“It’s not really feasible, but it’s an opening offer,” said Shira Efron, an analyst at the Israel Policy Forum, a New York-based research group. “It could force a discussion that bridges the two sides’ positions to extend the cease-fire for a couple weeks or more.”

Still, she said, it does not resolve the underlying differences between Hamas and Israel about the end of the war.

At a government meeting on Sunday, Mr. Netanyahu said the proposal included a temporary cease-fire during the Muslim fasting month of Ramadan and the Jewish holiday of Passover. Half of the remaining hostages in Gaza, he said, would be returned to Israel at the beginning of the temporary cease-fire and the other half would be repatriated at the end, if an agreement on a permanent cease-fire is concluded.

In the first phase of the three-stage deal agreed to in January, Hamas released 25 Israeli hostages and handed over the bodies of eight others in exchange for more than 1,500 Palestinians jailed by Israel. But without further planned exchanges of hostages and prisoners, Israel will have fewer incentives to keep the truce going.

On Sunday, Hamas dismissed the new proposal as “a blatant attempt to renege on the agreement and evade negotiations for its second phase.”

Hamas considers the idea of immediately giving up half of the hostages a nonstarter, but it could consider exchanging a small number of hostages or bodies for Palestinian prisoners, even without a commitment to the end of the war, analysts said. The hostages represent Hamas’s most powerful leverage, and every time it trades an Israeli captive for Palestinian prisoners, its negotiating hand is weakened.

Two Israeli officials, who spoke on condition of anonymity to discuss internal deliberations, acknowledged that Hamas would probably be willing to give up only a small number of hostages, or their remains, without guarantees for the end of the war. That dynamic, the officials said, may eventually make Israel choose between restarting a war to unseat Hamas or saving hostages still believed to be alive.

About 25 captives and the remains of more than 30 others are still in Gaza, according to the Israeli government.

“Israel stands on the horns of a dilemma,” said Yaakov Amidror, a retired major general who served as Mr. Netanyahu’s national security adviser.

On Sunday, Hazem Qassim, a spokesman for Hamas, said the militant group was insisting on negotiating the second phase because it wanted to prevent the resumption of the war and ensure Israel withdraws from Gaza.

“This is a fundamental position for the Hamas movement,” he told the Qatari-funded broadcaster Al Jazeera.

Both Israel and Hamas have sent negotiators to speak with Egyptian and Qatari mediators. But even as the diplomatic discussions continue, the two sides are preparing for the possibility of a return to war.

Hamas has been collecting unexploded bombs throughout Gaza and repurposing the explosives and their metal cases as improvised explosive devices, according to one member of the Qassam Brigades, Hamas’s military wing, who spoke on condition of anonymity to discuss sensitive details. The militant group has also been recruiting new members and replacing commanders killed in the fighting, the person said.

Israel has prepared extensively for a new and intense campaign in Gaza, according to Israeli officials. They said any new operations would include targeting Hamas officials who siphon off aid supplies meant for civilians, as well as destroying buildings and infrastructure used by the Hamas-run civilian government.

Such a plan has not yet been approved by the Israeli cabinet, the officials said, but they believe that only Mr. Trump could dissuade Mr. Netanyahu from renewed war.

While Israel and Hamas struggle over Gaza’s future, Palestinian civilians in the enclave, and the families of hostages, are facing an anxious period of limbo.

“They’re being left in a state of perpetual worry,” said Akram Atallah, a London-based Palestinian columnist originally from Jabaliya in the northern Gaza Strip. “If the war returns, they stand to lose the most.”

Patrick Kingsley and Ronen Bergman contributed reporting.