Under the terms of a proposed Gaza cease-fire and hostage release deal, Israel could see only around a dozen captives taken during Hamas' Oct. 7 attack returned home alive during the initial weeks of the agreement, according to several U.S. officials familiar with the ongoing negotiations.

The unfavorable return for Israel is viewed by the Biden administration as a major impediment to brokering an end to the conflict, and one of the factors fueling the renewed urgency of its campaign to push talks over the finish line.

The White House says negotiations resumed in Egypt this week, but so far mediators have been unsuccessful in their efforts to encourage Hamas to sign onto a bridging proposal that Israel has agreed to which aims to get talks back on track.

"We'll never give up on it, but the challenge is the longer this goes on, the more hostages will suffer -- possibly perish -- and the more other things happen that could make things impossible," Secretary of State Antony Blinken said during a visit to Israel on Monday.

Documents reviewed by ABC News detailing the framework for an agreement, which was partially outlined in a late May address by President Joe Biden, stipulate that Hamas must turn over 33 Israeli hostages in the 42 days after the deal is enacted -- including all living Israeli women, children, men over the age of 50, as well as wounded and ill male civilians under the age of 50.

However, in the event Hamas cannot provide 33 living hostages that fit the criteria, the proposed terms allow the militant group to fill the quota by turning over the remains of hostages in these categories.

Out of the 109 hostages still believed to be held in Gaza, Israeli authorities publicly say about 70 captives could still be alive. But four U.S. and Israeli officials say the actual number of living hostages is assessed to be about 50 and that only a fraction of them are either women or old enough to qualify as elderly under the terms of the proposal.

Israel has provided mediators with a list of all the individuals it wants Hamas to turn over in the initial weeks of a deal which includes civilian men under 50 that are believed to be alive but injured or ill. However, two U.S. officials familiar with negotiations anticipate that the militant group will be reluctant to hand over any of the younger men before reaching the second phase of the agreement regardless of their condition.

Since creating the framework, officials say Israel has added new conditions that Hamas rejects, including refusing to withdraw Israel Defense Forces from strategic areas of Gaza and insisting on screening Palestinians returning to their homes in the northern part of the enclave for weapons.

After initially denying that any changes were made to the proposal, Prime Minister Benjamin Netanyahu now admits to adding conditions he says are red lines and has reportedly expressed skepticism that a deal can be reached.

Several senior Israeli officials say Netanyahu is the driving force behind the new demands and have indicated the security benefits from the additional conditions are negligible when compared to the urgent need for a hostage deal and a cease-fire.

On Wednesday, Israeli Defense Minister Yoav Gallant visited IDF troops at the Philadelphi corridor between southern Gaza and Egypt and suggested there was no reason to continue occupying the area.

"The most important thing is to remember what the goals of the war are and to stick to them," he said. "The Rafah Brigade has been defeated."

"The Rafah Brigade has been defeated,” Gallant continued, referring to Hamas’ battalions that previously operated in the southern Gaza city.

Gallant has also described reaching a deal to release the hostages as "an urgent, moral imperative" in recent days -- a position that sparked public backlash from Netanyahu.

U.S., Israeli and Egyptian officials met on Thursday to discuss options for securing Gaza's border with Egypt and reopening the Rafah crossing, according to an official familiar with the plans.

National Security Council spokesperson John Kirby said on Friday that those conversations had already been "constructive" and denied that talks were at risk of breaking down.

"There's been press reporting out there that, you know, they're near collapse. That is not accurate. There has been progress made. We need now for both sides to come together and work towards implementation," he said.

But the shrinking incentive for Israel to shift its position was illustrated this week when IDF troops retrieved the bodies of six hostages in the Gaza Strip. Five of those recovered were already thought to be dead, but one -- 79-year-old Avraham Munder -- was presumed to be alive before his remains were found.

In previous rounds of the on-off talks, officials say Israel has pushed for Hamas to provide a list of the hostages that would be returned as part of a deal before striking an agreement, but the militant group has refused to do so.

In November 2023, Israel and Hamas reached an agreement to exchange at least 50 women and children held in Gaza for 150 Palestinian prisoners and truce lasting at least four days.

Hamas ultimately released more than 100 Israelis and foreign nationals, prolonging the pause in fighting, but officials say the group eventually refused to free any additional women and fired on Israel, which lead to the resumption of the war.