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

The United Nations General Assembly has passed a resolution calling on the Security Council to reconsider Palestinian membership in the U.N.

The motion passed overwhelmingly with 143 countries in favor, nine against and 25 abstaining.

"Granting Palestinian membership requires a recommendation from the Security Council. At the same time, the Assembly determines that the State of Palestine is qualified for such status and recommends that the Security Council 'reconsider the matter favourably,'" the U.N. said in a statement.

Despite broad support, the U.S. voted "no."

"President Biden has been clear that sustainable peace in the region can be achieved only through a two-state solution, with Israel’s security guaranteed, where Israelis and Palestinians can live side by side with equal measures of freedom and dignity. It remains the U.S. view that unilateral measures at the UN and on the ground will not advance this goal," Ambassador Robert Wood said in a statement Friday.

If approved, the new status would not take effect until the General Assembly opens in September.

-ABC News' Will Gretsky