In a joint statement shared by the U.S. State Department on Saturday, mediators from the U.S., Egypt and Qatar said they’re working to secure a cease-fire in Gaza and the release of the hostages, and urged Hamas and Israel "to finalize the agreement" that President Joe Biden outlined on Friday.

"These principles brought the demands of all parties together in a deal that serves multiple interests and will bring immediate relief both to the long-suffering people of Gaza as well as the long-suffering hostages and their families," the statement said. "This deal offers a roadmap for a permanent ceasefire and ending the crisis."

An Israeli government official said the Israeli Prime Minister’s Office wasn’t informed that President Joe Biden’s Friday remarks would include a permanent end to the fighting.

“If you listen to what Biden said, you come to the impression that we agree to a permanent cease-fire without, or, without our conditions being met, and that's not the case,” the official said.

The official said the Israeli Prime Minister’s Office, including Prime Minister Benjamin Netanyahu, was informed 30 minutes before that Biden was giving a speech. The official said Netanyahu was told that the speech would focus on the hostages, with no mention of implementing a permanent cease-fire.

A senior U.S. official said the Biden administration "engaged with the Israelis at multiple levels before the president delivered the speech." The official said the U.S. specifically engaged with the Israeli Prime Minister's Office and the Israeli war cabinet.

The official also pointed to when Biden said in his speech that it was a proposal that the Israelis agreed to. The official noted that Netanyahu himself said that it was an Israeli proposal.

An Israeli official and two other Israeli sources who are familiar with the matter told ABC News that what Biden presented is indeed the outline of the Israeli proposal. The sources said the proposal was approved by all war cabinet members, including Netanyahu, and sent to the mediators earlier in the week.

The sources added that there are “interpretation gaps,” referring to the way the outline was presented by Biden.

Biden presented on Friday what he said was an Israeli proposal to draw the war in Gaza to a close.

Biden said the deal had been transmitted to Israel and Hamas, through Qatar.

The president said the first phase would last for six weeks and include a "full and complete cease-fire, a withdrawal of Israeli forces from all populated areas of Gaza [and] release of a number of hostages." Palestinians would also be able to return to their homes and 600 trucks of humanitarian aid would be delivered to Gaza every day.

The second phase would include a release of all remaining living hostages and the temporary cease-fire would become a permanent cease-fire with Israel withdrawing all forces from Gaza "as long as Hamas lives up to its commitments," Biden said.

In phase three, there would be a major reconstruction of Gaza.

-ABC News’ Britt Clennett, Dana Savir and Michelle Stoddart

Right-wing Israeli Ministers Bezalel Smotrich and Itamar Ben Gvir spoke out on social media Saturday, both threatening to leave the government if the deal in its current form is accepted.

"I have now spoken with the Prime Minister and made it clear to him that I will not be part of a government that will agree to the proposed outline and end the war without destroying Hamas and returning all the abductees," Smotrich wrote.

This comes after the Hostage Families group released a statement urging Israel's parliament to embrace the deal.

Egyptian and Israeli officials will hold talks in Cairo on Sunday to discuss the reopening the Rafah border crossing between Gaza and Egypt, an Egyptian security official confirmed to ABC News.

The crossing has been closed since Israeli forces took control of the Gaza side during their offensive toward Rafah in May. Egyptian officials have said they will not reopen the crossing as long as Israel controls the Gaza side.

Egypt has, however, agreed to send aid trucks from its side of the border to be processed from the Israeli border checkpoint of Kerem Shalom.

-ABC News’ Ayat Al-Tawy