Lior Rudaeff, 61, an Israeli hostage taken by Hamas, has been declared dead, after being killed in the Oct. 7 attack, the Hostage Release Center said.

Rudaeff volunteered as an ambulance driver. He was married to his wife, Yaffa, for 38 years and had four children: Noam, Nadav, Bar and Ben.

"The Israeli government has a profound moral duty to pursue every avenue in the current negotiations to bring Lior home. He deserves a dignified burial in his homeland, alongside the 38 other hostages brutally murdered. The government must also secure the swift return of all living hostages, so they can begin the long road to healing and recovery," a spokesperson for the Hostage Families Forum said in a statement Tuesday.

The bodies of 38 hostages who have been declared dead continue to be held by Hamas.

-ABC News' Will Gretsky

The White House is watching the situation that is playing out in Rafah "very, very closely," but arguing that based on what Isarel has told them, this is not equivalent to the full-scale invasion the U.S. has been warning Israel against undertaking, U.S. National Security Council Coordinator John Kirby told reporters Tuesday.

"We've been told by our Israeli counterparts this operation last night was limited and designed to cut off from Hamas' ability to smuggle weapons and funds into Gaza. Our views on Rafah remain the same. We've repeatedly and consistently expressed those views about a major operation in Rafah in densely populated areas that would give higher risk to civilians and civilian casualties," Kirby said.

"President Biden made that very clear yesterday when he spoke to Prime Minister Netanyahu as I said yesterday, and we're going to be monitoring this situation very closely to see how it unfolds," Kirby said.

Kirby was also asked if he believes that Isarel was abiding by international law, given the two main arteries to get aid into Gaza have been choked off. Kirby sidestepped the questions, underscoring the U.S. desire to get more aid into Gaza and Biden's explicit ask of Netanyahu to make sure that happens during their call Monday.

-ABC News' Molly Nagle

Hamas' delegation has arrived in Cairo as cease-fire negotiations continue with mediators in Egypt and Qatar.

Hamas said Monday it had agreed to a cease-fire deal, but Israel said the agreement included "significant gaps" that needed to be worked out in negotiations.

-ABC News' Will Gretsky

The Biden administration is delaying shipments of ammunition that were expected to be sent to Israel, according to three U.S. officials.

The Biden administration has been trying to pressure Israel to hold off on a full-scale invasion of Rafah until it has a plan to protect the million civilians now living there.

The Air Force has been told by the White House’s National Security Council to pause shipments of ammunition deliveries specifically from Dover Air Force Base, according to a U.S. official. The NSC was behind the decision to pause the munitions, a second U.S. official confirmed.

The National Security Council did not immediately respond to a request for comment.

-ABC News' Martha Raddatz, Anne Flaherty, Matt Seyler and Ben Siegel

In a phone call Sunday with his Israeli counterpart, White House National Security Advisor Jake Sullivan expressed an "ironclad U.S. commitment" to Israel but also voiced the Biden administration's concerns about Israel's major military operations in Gaza, according to the White House.

During the call with Israeli National Security Advisor Tzachi Hanegbi, Sullivan reiterated President Joe Biden's "longstanding concerns over the potential for a major military ground operation into Rafah, where over one million people have taken shelter," according to a readout of the call that was released by the White House.

"He [Sullivan] discussed alternative courses of action to ensure the defeat of Hamas everywhere in Gaza," the readout said. "Mr. Hanegbi confirmed that Israel is taking U.S. concerns into account."

The White House said Sullivan also expressed condolences on Israel's Memorial Day, the first since Hamas' surprise Oct. 7 attacks on Israel. The Hamas attack killed 1,200 people, most of them civilians, according to Israel's Prime Minister's Office.

Sullivan and Hanegbi also reviewed discussions by officials on both sides of the war about alternatives for a Rafah invasion and agreed to plan an in-person meeting soon, according to the White House.

-ABC News' Michelle Stoddart