The Israeli military moved into eastern Rafah late Monday to begin a "precise counterterrorism operation" targeting Hamas, the Israel Defense Forces said Tuesday.

IDF troops also took "operational control" of the Gazan side of the border crossing between Gaza and Egypt, officials said in a statement.

The operation in Rafah overnight included air strikes and on-the-ground troops, the military said. About 20 "terrorists" were killed and three tunnels were located, IDF officials said.

"IDF ground troops are continuing to operate against Hamas terrorist operatives and infrastructure in the area of the Rafah Crossing in eastern Rafah," the statement said.

-ABC News' Jordana Miller, Jessica Gorman and Kevin Shalvey

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

An independent, self-formed task force of former State Department officials and international law experts say the U.S. "shirked" its legal obligations with Friday's National Security Memorandum-mandated report that examined Israel's use of U.S.-supplied arms.

"The Administration must come to terms with the fact that what it faces in its lethal military assistance to Israel's operations in Gaza - and beyond - is not a policy problem: it is a legal problem," the officials said Friday, adding, "The Administration’s report is at best incomplete, and at worst intentionally misleading in defense of acts and behaviors that likely violate international humanitarian law and may amount to war crimes. Once again, the Biden Administration has stared the facts in the face - and then pulled the curtains shut."

More than 34,000 Palestinians have died, including 14,000 children, the officials said, noting that "most of them [were] killed by U.S. munitions."

-ABC News' Christopher Boccia