Israeli war cabinet members could oppose a suggested Gaza ceasefire agreement, fearing it would not be implemented beyond its first phase.
The US proposal would see about 40 out of 129 hostages held in Gaza by Hamas and other groups released in the initial phase, although 34 captives are believed to be dead.
The war cabinet, which does not include Prime Minister Benjamin Netanyahu's extreme-right allies Bezalel Smotrich and Itamar Ben-Gvir, is in favour of reaching an agreement regarding the hostages even if it is costly for Israel.
But they say Israel should possibly demand a one-phase agreement covering all captives who remain alive in Gaza.
A previous week-long truce happened over four months ago in November, they said. The pause in fighting included a hostage exchange but left dozens of hostages in the hands of Hamas and other Palestinian groups.
The cabinet members believe it would be more difficult to realise an agreement on a second stage than on the first, saying this would involve troops remaining in Gaza.
Talks between Hamas and Israel dragged on, with the Palestinian Islamist group accusing Israel of intransigence while at the same time studying a new ceasefire proposal
Israeli media reported on Tuesday that Israel was insisting on having a veto allowing it to prevent some Palestinians from being freed as part of a hostage exchange with Hamas.
It comes as the threat of famine looms in Gaza, where Israeli forces have killed tens of thousands of people while devastating the territory's infrastructure. Hospitals, ambulances, and residential buildings have come under attack.
33,360 people have been killed and 75,993 injured, according to the enclave's health ministry.
The International Court of Justice (ICJ) found in January that Israel was plausibly breaching the UN's Genocide Convention in Gaza.
Israeli Prime Minister Benjamin Netanyahu said on Tuesday: "Our goal is to return the hostages. All the [captured] lookouts, and not only them, all of them together.
"We'll bring them all home."
However, extremist ministers Ben-Gvir and Smotrich have been pressuring him not to sign a ceasefire agreement and threatened to leave the government if he does not go ahead with an invasion of Rafah, where 1.4 million Palestinian civilians have fled from Israeli attacks elsewhere in Gaza.
Netanyahu insists the attack on Rafah will take place and a date had been set for it, but has refused to disclose this date.
The prime minister has reportedly tried to convince Smotrich to back a deal with Hamas but not Ben-Gvir, who he believes will be against the agreement under any circumstance.
A source close to the Likud Party leader said he can accept Ben-Gvir's Jewish Power Party and Smotrich's Religious Zionism faction going against the agreement as it does not need their backing to be adopted.
"The only thing that worries Netanyahu is their departure from the government. And our assessment is that this isn't really on the agenda," the source said.
Smotrich had initially made his threat over the fact there had been no full security cabinet meeting to talk about the agreement, though a session was held on Tuesday.
