Israel and the United States have both blamed Hamas for failing to agree to a proposal that would free about half of the remaining Hamas-held Israeli hostages, estimated to total around 100, during a six-week pause in fighting that would also facilitate a significant increase in humanitarian aid entering Gaza.

But Hamas is holding out for Israel to agree to a more permanent end to the fighting. The government of Prime Minister Benjamin Netanyahu has adamantly refused, saying that it is determined to destroy the militant group. He has vowed that Israel will attack Hamas fighters said to be hiding among the roughly 1.5 million Palestinians now squeezed into the southern part of the enclave.

“The ball is in both Hamas’s and Israel’s court,” said a person familiar with the negotiations, who spoke on the condition of anonymity to discuss the sensitive talks.

Burns, a former career diplomat with decades of experience in sensitive diplomacy, touched down for talks in all three capitals between Thursday and Saturday, which included a meeting in Amman with David Barnea, head of Israel’s Mossad intelligence agency. He has been the Biden administration’s point person in negotiations between Israel and Hamas, traveling to Doha in late November for a final shove that achieved an earlier week-long pause in the fighting and an initial release of hostages.

Since the current talks got underway in January, Burns has traveled to Paris, Cairo, Tel Aviv and Doha, where Hamas’s political leadership is based. Barneahas been at many of those meetings, and this time he traveled to Amman. U.S. and Arab officials say the two spy chiefs are able to speak without the burden of political considerations, in a manner sometimes difficult in conversations between the top levels of their respective governments.

It was not clear whether any progress had been made before Burns left Amman to return to Washington on Saturday. A Mossad statement released by Netanyahu’s office said “contacts and cooperation with the mediators continue all the time in an effort to narrow the gaps and reach agreements,” Reuters reported from Jerusalem late Saturday.

During weeks of negotiations in which the United States, Qatar and Egypt have served as intermediaries, the warring sides have indirectly debated a number of issues, among them a Hamas demand for specific amounts of aid to go to specific places in Gaza, and Israel’s insistence on a complete list of the remaining hostages and how many of them are still alive.

Israel, while agreeing to reposition its troops in southern Gaza during the six-week pause, wants to retain freedom of military movement in the north. Hamas wants guarantees that all Gazans from the largely evacuated north can return there — with the provision of food, water, medical supplies and shelter. There also have been disagreements over how many Palestinians held in Israeli prisons, and which ones, will be released in exchange for the hostages.

But these issues are not “sticking points,” said the person familiar with the talks. “If you deal with the main one — a cease-fire — they can be sorted out.” The issue of a temporary pause vs. a permanent end to the fighting led to last weekend’s breakdown of talks held in Cairo.

For its part, the Biden administration is under increasing pressure to get a deal done now. It envisions what a senior administration official called a “three-phase deal,” beginning with the initial six–week pause in fighting, the concurrent release of all women, children, elderly and injured hostages, and a guaranteed increase in aid. That first tranche of the hostage release would include at least one of six American citizens being held, Hersh Goldberg-Polin, who was severely wounded and lost an arm in the Oct. 7 Hamas attack that killed around 1,200 people in southern Israel. The attack led to the ongoing Israeli military offensive in which more than 30,000 have been killed in Gaza, President Biden said on Thursday.

“All of that stuff has been negotiated,” the official said. Release of the remaining hostages, including captured members of the Israel Defense Forces, and a continuation of the cease-fire “can be extended into a second phase.”

The U.S. view is that agreement on the first phase is “a step in the right direction — you get 45 days, you give people relief during Ramadan,” and secure the release of many of the hostages, the person familiar with the talks said. Once the initial period of calm has been established, the hope is that both sides will feel pressure to agree to something more permanent.

But Hamas, which sees its leverage diminished with every hostage released, has not yet signed off on a stand-alone first phase to which the administration says Israel has agreed. Israel has said it intends to return to the fight as soon as it has all of the captives back. Neither side trusts any guarantees signed by the other.

Stopping the fighting during Ramadan is seen as a significant goal. One of the five pillars of Islam, observed by Muslims worldwide, the month is marked by fasting, introspection, prayer and ideally a family gathering each evening after sundown. The last day of the fast, Eid al-Fitr, is one of Islam’s two official holidays.

The holiday is expected to sharply increase tensions in Gaza, where much of the displaced population is living in squalid tent and plastic sheeting camps in the south. An estimated 300,000 Palestinians remain in the north, which has been reduced to large swaths of rubble by Israeli air and ground strikes. Reports of mass starvation in the north, unreachable by most aid convoys, spurred Biden to authorize food airdrops and the U.S. military construction of a temporary port on Gaza’s Mediterranean coast to allow mass deliveries of assistance.

But as the talks advance with the informal Ramadan deadline looming, officials across the region are concerned about the possibility of other violence. In East Jerusalem, members of Netanyahu’s right-wing coalition government have threatened to limit access to the al-Aqsa Mosque, one of Islam’s holiest places and a traditional flash point for Israeli-Palestinian clashes. Hamas political leader Ismail Haniyeh has called for Palestinians to march there on the first day of the fast. In the West Bank, clashes between Israeli settlers and security forces and Palestinians are expected to increase.

“It’s always a very volatile period,” the Biden administration official said. “I’m not going to make a prediction” on when a deal might be reached, but “obviously, we recognize that extremists could try to use Ramadan to spark something that would be deeply unfortunate in that holy month.”

Biden and former president Donald Trump had dueling rallies Saturday night. Trump falsely claimed that Biden “announced a plan to send our brave U.S. military men and women into Gaza to resupply the terrorists of Hamas.” During the State of the Union Thursday, Biden said that the U.S. military would construct a temporary pier on the coastline of Gaza to provide a new route for humanitarian aid to reach desperate civilians.

U.S. officials said construction of the facility will involve the presence of U.S. military personnel on military vessels offshore from Gaza, but will not require U.S. military personnel to install the pier ashore or to offload the aid. The aid will be distributed inside Gaza by the United Nations and other humanitarian personnel.