The White House expects the Kerem Shalom crossing to reopen on Wednesday, but are working to move up that timeline to get vital humanitarian assistance into northern Gaza even sooner.

“Israel has committed to reopening the Kerem Shalom,” White House press secretary Karine Jean-Pierre told ABC News. “We've been told that that's going to happen tomorrow. We are working to move that up.”

“Fuel deliveries through Rafah should also commence tomorrow. That's what we've been told. And we believe Rafah border crossing should also be quickly reopened for the movement of humanitarian assistance. We are insisting that there should be no disruptions of humanitarian assistance flows," Jean-Pierre said.

Jean-Pierre would not say if the U.S. still expects the Israelis to alert them when they do plan to go ahead with a full-scale operation in Rafah, but reiterated that they do not support the IDF carrying that out.

"We do not want to see a major operation happening in Rafah. We want to see a plan, a comprehensive plan, on how Israel plans to protect the 1.5, more than 1 million civilians that are currently seeking refuge in Rafah," Jean-Pierre said.

-ABC News' Mary Bruce and Justin Gomez

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

More than 300,000 people have fled Rafah in the week since Israel issued a partial evacuation order, the United Nations agency operating in Gaza said on Sunday.

The U.N. Agency for Palestine Refugees called the evacuation "forced and inhumane."

"There is nowhere safe to go," the agency said on social media, repeating the phrase three times for emphasis.

The Israeli military late Saturday called again for civilians to evacuate from much of the eastern part of the city, which is in southern Gaza.

Israel Defense Forces entered Rafah last week, in what they called a "precise" operation ahead of potential invasion.

"Prior to our operations we urge civilians to temporarily move towards humanitarian areas and move away from the crossfire that Hamas puts them in," the Israel Defense Forces said on a post on Telegram. "Our war is against Hamas, not against the people of Gaza."

-ABC News' Dragana Jovanovic and Kevin Shalvey