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

A new report from the United States Departments of State and Defense said that given "Israel's significant reliance on U.S.-made defense articles" it is "reasonable to assess" that some have been used in instances "inconsistent" with Israel's obligation under international law.

The assessment, which came as part of a report ordered by President Joe Biden, examined the use of U.S.-supplied arms in active conflict zones, and also said that the U.S. could not compile enough evidence to prove that the country used American defense articles in violation of international humanitarian law.

"We do not have complete information to verify whether US defense articles covered under NSM-20 were specifically used in actions that have been alleged as violations of IHL (international humanitarian law) or international human rights law during the period of the report," the document says. "The nature of the conflict in Gaza makes it difficult to address or reach conclusive findings on individual incidents."

While the report said the Israeli government has "the knowledge, experience, and tools" to implement "best practices for mitigating civilian harm," the high number of civilian casualties "raise substantial questions about whether the IDF is using them effectively in all cases."

Additionally, the report noted that Israel did not cooperate with U.S. efforts to ramp up humanitarian aid and distribution in Gaza during the initial months of the conflict, but that officials have observed improvements.

-ABC News' Shannon Crawford and Anne Flaherty