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

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