Israel-Gaza updates: Crossing closed to aid after rockets fired from Rafah: IDF

United Nations Secretary-General Antonio Guterres provided an update on the hunger situation in Gaza and said there has been " incremental progress recently" in preventing a famine.

However, Guterres warned there is still more work needed, "including the promised opening of the two crossing points between Israel and northern Gaza so that aid can be brought into Gaza from Ashdod, Port and Jordan are still obstacles including a lack of security for aid convoys."

"Humanitarian convoys, facilities and personnel and people in need must not be targets," he added.

Guterres said that civilians were dying from hunger and disease every day and pressed for a successful negotiation for a cease-fire.

"I strongly encourage the government of Israel and Hamas leadership to reach now an agreement. Without that I fear the war with all its consequences, both in Gaza and across the region, will worsen exponentially," he said.

-ABC News' Will Gretsky

The Jordanian government released a readout of the meeting between King Abdullah II and Secretary of State Antony Blinken on Tuesday.

Abdullah "warned of the danger of any military operation in Rafah, stressing that the catastrophic effects of the war in Gaza could spread to areas in the West Bank, Jerusalem, and the entire region," according to the readout.

The king also called for more aid to Gaza "through all available means," the readout said.

"The King said supporting UNRWA is crucial to enable it to cover the basic needs of nearly 2 million Palestinians in Gaza, as well as other Palestinian refugees in its areas of operation," the readout said.

The U.S. and some other countries pulled funding for UNRWA, the United Nations Relief and Works Agency for Palestine Refugees in the Near East, after Israel said several staffers took part in Hamas' Oct. 7 attack.

-ABC News' Cindy Smith

Israel will not send a delegation to Cairo for negotiations on a cease-fire deal until Hamas provides an answer to the proposal Israel has offered them, an Israeli source told ABC News.

-ABC News' Jordana Miller

White House national security spokesman John Kirby declined to comment about Israeli Prime Minister Benjamin Netanyahu's latest comments about a possible operation into Rafah during a news conference Tuesday.

"Our position on Rafah is absolutely the same. We don't want to see a major ground operation in Rafah. Certainly, we don't want to see operations that haven't factored in the safety and security of those 1.5 million folks trying to seek refuge down there," Kirby said.

He also declined to say if Netanyahu has shared his plans to enter Rafah with or without a deal directly with the United States.

"They understand our concerns, and those concerns have not changed," he said, adding that they have not seen a credible plan yet from Israel to take civilian safety into consideration though conversations continue.

Kirby reiterated that the new cease-fire deal is on the table, which would give a six-week pause in fighting and help get the hostages home.

Kirby downplayed expectations, but stressed "time is of the essence."

"I wouldn't say we're overly confident. I would say we're being very pragmatic about this," he said.

-ABC News' Molly Nagle