They had reason to believe significant progress had been made: Mr. Biden said he was sharing the outline of an Israeli offer, one that included language that was closer to Hamas’s demand of a permanent cease-fire, and Hamas initially responded to the American president’s remarks saying it viewed them positively.
Now, four days later, the families of hostages have grown increasingly concerned that the latest proposal will end in failure. Far-right members of Israel’s coalition have threatened to take down the government if Israel agrees to end the war without toppling Hamas, and Hamas has not formally consented to the proposal shared by Mr. Biden.
The latest impasse has reinforced the feeling among relatives of hostages that they have been on a seemingly never-ending roller-coaster ride, a vicious cycle of raising their hopes for a deal only to be disappointed time and time again.
“It’s incredibly frustrating to have this yo-yo experience once again,” said Lee Siegel, the brother of Keith Siegel, a 65-year-old Israeli-American hostage. “Every day that goes by it becomes exponentially more difficult to continue with hope.”
Advocates for the release of hostages have warned that the conditions for them have been growing more precarious each day, especially for older adults and people who are ill. Without an agreement, they worry that the majority of the hostages in Gaza won’t return to Israel alive.
“Everyone is terrified that they’ll receive a call from the authorities informing them that their family member was found dead,” said Gili Roman, the brother of Yarden Roman-Gat, who was released from captivity in November. Ms. Roman-Gat’s 40-year-old sister-in-law, Carmel Gat, was still being held in Gaza, according to Israel.
On Monday, the Israeli army announced that it had concluded that four hostages had died in the Khan Younis area months ago.
The hostage families’ mounting concerns come as Prime Minister Benjamin Netanyahu mulls whether to push ahead with the proposal on the table in the face of countervailing pressures: As the international community demands a cease-fire and Israel isolates itself further, his domestic right-wing allies, whose support he has relied on to remain in power, reject any agreement that would lead to an end of the war without eliminating Hamas.
Another layer of pressure has come from the families of hostages, but some have lost hope that demonstrations in Israel will compel Mr. Netanyahu to agree to a deal and believe only unrelenting American pressure will make that happen.
“The United States should not leave Netanyahu for a second until he signs on to an agreement. Not a second,” said Gilad Korngold, the father of Tal Shoham, a 39-year-old hostage from northern Israel. “That’s what’s needed.”
Mr. Korngold said that three members of his family were killed on Oct. 7, and six others who had been abducted were released during a short-lived cease-fire in late November.
While Israeli officials have said the proposal Mr. Biden presented generally aligned with a new offer approved by Israel’s war cabinet, Mr. Netanyahu has rejected ending the war without destroying Hamas’s governing and military capabilities.
Mr. Biden said if Hamas abided by the proposal he outlined, that would ultimately lead to the “cessation of hostilities permanently.”
Mr. Korngold, however, said that the American government also needed to apply unyielding pressure on Hamas through Qatar to force them to approve an agreement.
“Hamas is not saying no but it’s also not saying yes,” he said. “It also has to get the message.”
Hamas officials have claimed that it was up to Israel to agree to a cease-fire and dismissed American calls on the group to approve the proposal.
As time passes, Mr. Siegel said he was becoming particularly fearful for his brother because he was diagnosed with high blood pressure last year.
“His absence weighs on us every minute, every hour, every day,” he said. “Each day he isn’t here could be his last day.”
