The latest round of talks between Israel and mediators began on Thursday and was expected to resume on Friday after noon prayers.
Mediators have said they are briefing Palestinian militant group Hamas, which is not taking direct part in the talks, on developments.
"This is vital work. The remaining obstacles can be overcome, and we must bring this process to a close," US national security spokesperson John Kirby told reporters at the White House.
Months of on-off talks have so far failed to overcome fundamental divisions, with Israel saying peace will only be possible if Hamas is destroyed, and Hamas saying it will only accept a permanent, rather than temporary, ceasefire.
Israel issued new evacuation orders for people to leave areas previously designated safe zones. (AP PHOTO)
Overnight, Israeli forces pounded targets across tiny, crowded Gaza and issued new orders for people to leave areas it had previously designated as civilian safe zones, saying Hamas had used them to fire mortars and rockets at Israel.
The conflict began on October 7 when Hamas fighters rampaged into Israel, killing about 1200 people and seizing about 250 hostages, according to Israeli tallies.
Israel's military campaign in Gaza has reduced much of the territory to rubble and killed more than 40,000 Palestinians, mostly civilians, according to Palestinian health authorities.
Israel says it has eliminated 17,000 Hamas fighters.
In a statement late on Thursday, Hamas politburo member Hossam Badran said Israel's continuing operations were an obstacle to progress on a ceasefire.
The Israeli delegation includes spy chief David Barnea, head of the domestic security service Ronen Bar and the military's hostages chief Nitzan Alon, defence officials said.
The White House sent CIA Director Bill Burns and US Middle East envoy Brett McGurk.
Qatari Prime Minister Sheikh Mohammed bin Abdulrahman Al Thani and Egypt's intelligence chief Abbas Kamel are also taking part.
The talks aim at halting the Israel-Hamas war and securing the release of hostages. (AP PHOTO)
The negotiations are taking place in the shadow of a feared regional escalation, with Iran threatening to retaliate against Israel after the assassination of Hamas leader Ismail Haniyeh in Tehran on July 31.
With US warships, submarines and warplanes dispatched to the region to defend Israel and deter potential attackers, Washington hopes a ceasefire agreement in Gaza can defuse the risk of a wider war.
Republican US presidential candidate Donald Trump said on Thursday he had told Israeli Prime Minister Benjamin Netanyahu during their last meeting in July to quickly end the Gaza war, but the former president also criticised ceasefire demands.
"He knows what he's doing, I did encourage him to get this over with," Trump told reporters at a news conference on Thursday.
"It has to get over with fast. ... Get your victory and get it over with. It has to stop, the killing has to stop."
Netanyahu's office and Trump both separately denied on Thursday an Axios report that said they had spoken the previous day about Gaza ceasefire and hostage release talks.
The White House, meanwhile, said attacks by Israeli settlers on Palestinian civilians in the West Bank were "unacceptable and must stop" after dozens of settlers assaulted a village, killing at least one person.
