Cease-fire talks are now moving forward at a working group level in Cairo over the next few days to hammer out specifics, according to National Security Communications Adviser John Kirby.

This weekend's strikes by Israel and Hezbollah have "not affected the actual work on the ground by the teams trying to get this cease-fire deal in place," Kirby told reporters Monday.

Kirby also rejected any suggestion that talks broke down this weekend, instead saying they were "constructive" enough to work on "finer details" at lower levels.

"There was no breakdown," he said. "They made enough progress that they were willing to, or needed to transition to a working group level so you didn't need the mediators all there and the leadership there."

Brett McGurk, a top senior adviser on the Middle East at the White House, stayed in Cairo an extra day to kick off the meetings and is still there, Kirby said, adding that all parties are being represented in these discussions.

"One issue that will be for the working groups to flesh out is the exchange of hostages and prisoners that Israel's holding -- what that exchange looks like, how many, some of the details of exactly who will be released on either side and at what pace, those kinds of things," Kirby said.

-ABC News' Justin Gomez

Al-Aqsa Hospital in Deir al Balah is still operating, despite new temporary evacuation orders from Israeli forces to leave the surrounding area near the hospital.

Out of the 650 patients in Al-Aqsa Hospital, only 100 remain in the hospital that are being treated, the Hamas-run Gaza Ministry of Health said.

The Israel Defense Forces acknowledged they have been "operating in recent days in the Deir al Balah area," but they said the evacuation orders did not include "the hospitals and medical facilities in the area," in response to an inquiry from ABC News.

Three out of 18 water wells are still functioning in Deir al Balah due to "ongoing military operations," the U.N. Agency for Palestinian Refugees said in a post Monday.

-ABC News' Diaa Ostaz and Jordana Miller

The World Food Programme, the U.N.'s worldwide food assistance program, is being "severely hampered" by the "intensifying conflict" in Gaza.

The agency said border crossings have been limited and roads in Gaza have become so unusable that urgent repairs are needed in order to transport basic needs, like food and medicine.

"Transporting food, water, medicine and hygiene equipment is critical for the survival of communities in Gaza today and will be needed for months to come," Antoine Renard, the country director for Gaza, said in a statement. "Roads are part of this lifeline."

Israeli forces issued an evacuation order in the vicinity of the Al Aqsa Hospital, Deir Al Balah, in central Gaza, urging people to flee, according to a statement from Doctors Without Borders Sunday.

"An explosion approximately 250 meters away triggered panic with many choosing to leave the hospital," the organization said.

Of the approximately 650 patients in the hospital prior to the explosion, only 100 remain, with seven in the intensive care unit, according to the Palestinian Ministry of Health.

Doctors Without Borders is considering suspending wound care for the time being, while trying to maintain lifesaving treatment, according to the statement.

"This situation is unacceptable," the organization said. "Al Aqsa has been operating well beyond capacity for weeks due to the lack of alternatives for patients. All warring parties must respect the hospital, as well as patients' access to medical care."