With CIA Director Bill Burns and White House Middle East adviser Brett McGurk working to advance a cease-fire and hostage release deal in the Middle East, two U.S. officials familiar with the negotiations said they see more flexibility from Hamas than they did previously.

The officials assessed this is coming from Hamas leader Yahya Sinwar; however, they stopped short of saying they felt a breakthrough was in the offing.

The officials also said they believe Israeli Prime Minister Benjamin Netanyahu is still committed to achieving a deal.

The American delegation headed by Burns arrived in Cairo earlier Monday to continue Gaza cease-fire talks, Egyptian state TV reported.

-ABC News' Shannon Kingston

A new evacuation order in Gaza will "further impede delivery of very limited life-saving care," the World Health Organization said Monday.

The Israel Defense Forces issued an evacuation order in Gaza City on Monday, telling people to go to designated safe zones.

Al-Ahli and Patients' Friendly hospitals are now "out of service," with some patients referred to hospitals that are experiencing shortages of beds and medical supplies, according to WHO Director-General Tedros Adhanom Ghebreyesus. Multiple health care centers are also located within the evacuation zone, while three hospitals are close by, he said.

"These key hospitals and medical facilities could quickly become non-functional due to hostilities in their vicinity or obstruction to access," he said in a statement.

The IDF said Monday it launched a "counterterrorism operation in Gaza City targeting Hamas and Islamic Jihad infrastructure."

"Civilians were warned and a route was opened for their safe evacuation," the IDF said.

Israeli forces claimed Monday they have killed another Hezbollah leader in southern Lebanon.

An Israel Defense Forces aircraft "struck and eliminated the Hezbollah terrorist Mustafa Hassan Salman in the area of Qlaileh," the military said in a statement.

The IDF accused Salman of participating in the "planning and execution of numerous terror attacks" against Israel.

Last week, the IDF said it assassinated Muhammad Ni'Mah Nasser, one of the most senior Hezbollah commanders killed since the group began firing at Israel in solidarity with Gaza on Oct. 8.

Attacks between Israel and Hezbollah forces continued Monday.

-ABC News' Jordana Miller and Ghazi Balkiz

The Hostages and Missing Families Forum has asked Israeli Prime Minister Benjamin Netanyahu to delay his trip to the United States at the end of the month and stay in Israel to "prioritize finalizing a hostage release deal."

Netanyahu is currently scheduled to address a joint session of the U.S. Congress on July 24.

"For 9 months, 120 innocent men, women, children, and elderly have been held captive in Gaza by Hamas," the Hostages Families Forum said in a statement on Monday. "These hostages endure unimaginable horrors daily. Every moment in captivity is a risk to their lives and well-being."

-ABC News' Jordana Miller