Qatar will send a delegation of mediators to Cairo on Tuesday to work on negotiations for a cease-fire deal, according to a spokesperson for the Qatari Foreign Ministry.

In a statement, the Qatari Foreign Ministry said they hope "talks will culminate in reaching an agreement for an immediate and permanent ceasefire in the Gaza Strip and the exchange of prisoners and detainees, and the sustainable flow of humanitarian aid into all areas of the Strip."

As cease-fire talks continue, IDF spokesperson Daniel Hagari again warned Palestinians in the eastern part of the city to evacuate.

"Tonight, we also call upon those staying in specific areas which we have communicated and defined through every means -- radio, media, internet, and flyers -- in eastern Rafah, to move towards the expanded humanitarian area in Al-Mawasi and Khan Yunis, where they will receive full humanitarian aid and where water, food, medical equipment, and shelter will be provided,” Hagari said.

Israeli Minister Benny Gantz said in a statement Monday the current proposal put forth by Hamas "has significant gaps.”

"Despite this, we continue to turn over every stone and a delegation will go to Cairo," he added.

Gantz said “every decision will be brought to the war cabinet,” which is made up of Israeli Prime Minister Benjamin Netanyahu, Israeli Defense Minister Yoav Gallant and himself.

-ABC News' Dana Savir

National Security Council spokesperson John Kirby told reporters that the U.S. is reviewing the Hamas response to a cease-fire proposal.

Kirby did not reveal many specifics about where negotiations stand, but said they’ve reached a “critical stage.”

“I don't know that it gets any more sensitive than right now,” Kirby said. “And the worst thing that we can do start speculating about what's in it.”

-ABC News' Molly Nagle

Hamas says negotiations with Israel for a permanent cease-fire have fallen through after Israel rejected its proposals, including demands for a permanent cease-fire, complete withdrawal of Israel's forces from Gaza, the return of displaced people and a prisoner exchange.

"Israel's rejection of the mediators' proposal, through the amendments it made to it, has brought things back to square one," Hamas said in a statement Friday.

Hamas said Israel bears "the full responsibility" for the failure to reach an agreement.

"The enemy's army's attack on Rafah and the direct occupation of the crossing immediately after Hamas announced its agreement to the mediators' proposal confirms that the occupation is evading reaching an agreement. Netanyahu and his extremist government are using negotiations as a cover for the attack on Rafah and the occupation of the crossing, and continuing the genocide against our people," Hamas said.