A spokesperson for the Hostages and Missing Family Forum, the organization representing most of the families, told NBC News that Israeli Prime Minister Benjamin Netanyahu had invited family representatives for a meeting on Tuesday in conjunction with the hostage deal.

The current proposal also calls for Israel to withdraw from populated areas of Gaza to the Israeli border. 

Washington is looking to secure a ceasefire deal before President-elect Donald Trump takes office next Monday, Jan. 20.

Trump has warned that “all hell will break out in the Middle East” if the remaining hostages, who have been held in Gaza after being taken captive in the Hamas-led Oct. 7, 2023, attacks, are not released before he assumes office.

A senior Arab diplomat directly involved in the talks told NBC News that there was “very real” headway being made. “We made very good progress,” they said.

The official said Trump’s Middle East envoy Steve Witkoff had played a critical role in advancing talks over the weekend, while one of the U.S. officials close to the negotiations said Witkoff and White House coordinator for the Middle East and North Africa, Brett McGurk, have been “working seamlessly” on the effort.

Witkoff, the U.S. official said, “has been very helpful in particular ironing out a couple of issues.” They did not expand on what those issues were, however.

The three-phase agreement is described as “very complex,” the U.S. official added, but noted it was “a massive deal that can end the war in Gaza at last and help reshape the Middle East.” 

The progress made in recent days follows repeated assurances from Washington since the Oct. 7 attacks, with the Biden administration undertaking frequent efforts to secure a deal between Israel and Hamas, with both sides trading blame for delays in progress.

It also follows a phone call Sunday between Biden and Netanyahu, in which the president “stressed the immediate need for a ceasefire in Gaza and return of the hostages” and a “surge in humanitarian aid.”

“Israel very much wants to release the hostages and is working hard to reach a deal,” Israeli Foreign Affairs Minister Gideon Sa’ar said Monday in a statement following a meeting with his Danish counterpart, Lars Rasmussen, in Jerusalem.

“There is progress in the negotiations,” Sa’ar said. “I thank our American friends for their help.”

Hamas said in a statement that members of its leadership council had “reviewed the progress made in recent days in Doha, stressing that things are going well, and that Hamas is dealing with these efforts and developments positively.”

In an earlier statement the militant group said it had stressed “its keenness to reach an agreement to stop the war on Gaza.”

Israel launched its assault on Gaza after Hamas led the Oct. 7 attacks, in which some 1,200 people were killed and around 250 taken hostage, according to Israel. The attacks marked a major escalation in a decadeslong conflict.

More than 46,500 people have been killed in Gaza since Israel launched its offensive in the enclave, according to local health officials, although the death toll is thought to be higher.