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

Although the Biden administration said it expected the Kerem Shalom and Rafah gates to be at least partially opened for humanitarian aid by Wednesday, the State Department was much less optimistic. However, the U.S. vowed it would continue to press Israel to get both portals up and running again.

While Kerem Shalom reopened Wednesday, aid had not been allowed to pass through because of "logistical and security concerns on the ground," State Department spokesperson Matt Miller said during a briefing Wednesday. He also said fuel had not been able to enter Gaza through Rafah.

"We think it's critical that Rafah be open for fuel because without fuel deliveries into Rafah, humanitarian assistance that comes in through Kerem Shalom can't actually be delivered. The two are critically linked," Miller said. "We're continuing to push for that to happen."

Miller didn’t provide any updated timeline on when the U.S. expected the crossings would be functional again, but said it was imperative that Israel act to increase the amount of aid allowed into Gaza following what he described as a "fairly dramatic dip" this week.

"When it comes to the level of humanitarian assistance, the results as they are today are just unacceptable. There's no other way to put it," Miller lamented.

He continued, "We are making quite clear to the government of Israel that we need to see those decreases reversed. We need to get back to where we were, and we need to see the levels continue to increase and see those sustained, and that is why we need to see Kerem Shalom and Rafah opened."

-ABC News' Shannon Crawford