Officials from the Islamist group and Israel have been holding talks with Qatari and Egyptian mediators in the most intensive effort for months to reach a ceasefire in the Gaza Strip.
The outgoing US administration has called for a final push for a deal before Joe Biden leaves office, and many in the region now view Trump's inauguration as an unofficial deadline.
But with the clock ticking, both sides accuse the other of blocking a deal by adhering to conditions that torpedoed all previous peace efforts for more than a year.
Hamas says it will free its remaining hostages only if Israel agrees to end the war and withdraw all its troops from the enclave.
Israel says it will not end the war until Hamas is dismantled and all hostages are free.
"Hamas is the only obstacle to the release of the hostages," the director general of Israel's foreign ministry, Eden Bar Tal, told a briefing with reporters, saying Israel was fully committed to reaching a deal.
Israel's military assault followed a Hamas attack in October 2023 that killed 1200 people. (AP PHOTO)
Hamas official Osama Hamdan, who held a news conference in Algiers, said Israel was to blame for undermining all efforts to reach a deal.
While he said he would not give details about the latest round of negotiations, he reiterated the Hamas conditions of "a complete end to the aggression and a full withdrawal from lands the occupation invaded".
Commenting on Trump's threat that there would be "hell to pay" unless all hostages were freed before the inauguration, Hamdan said: "I think the US president must make more disciplined and diplomatic statements."
Israel has sent a team of mid-ranking officials to Qatar for talks brokered by Qatari and Egyptian mediators.
Some Arabic media reports said David Barnea, the head of Mossad, who has been leading negotiations, was expected to join them.
The Israeli prime minister's office did not comment.
In one notable step towards a deal, a Hamas official told Reuters on Sunday the group had cleared a list submitted by Israel of 34 hostages who could be freed in the initial phase of a truce, alongside Palestinian prisoners held by Israel.
The list included female Israeli soldiers plus elderly, female and minor-aged civilians.
Israeli Prime Minister Benjamin Netanyahu's office said Israel had so far received no confirmation about whether those on its list were still alive.
Nearly 46,000 Palestinians have been killed in Israel's assault on Gaza, according to health officials in the enclave.
The assault was launched after Hamas fighters stormed Israeli territory in October 2023, killing 1200 people and capturing more than 250 hostages, according to Israeli tallies.
Israeli military strikes killed at least 10 Palestinians across the Gaza Strip on Tuesday, medics said, as the health ministry in the Hamas-run territory urged international donors to immediately provide fuel to maintain medical services.
One of those strikes killed four people in a house in Beach camp in Gaza City while the remaining six were killed in separate strikes across the enclave, medics said.
The health ministry said hospitals were running out of fuel to operate generators and maintain medical services across Gaza because of Israeli restrictions.
Israel has repeatedly said it facilitated the delivery of fuel and medical supplies to hospitals in the enclave, even in areas where forces have active operations.
On Tuesday, the military said 240 Palestinians its forces had detained in a raid on Kamal Adwan Hospital in northern Gaza last month had provided "substantial intelligence".
The military released footage of the interrogation of a purported Hamas militant who detailed how militants "operated from the hospital area" and transferred weapons to and from it.
Hamas and the health ministry deny any armed presence at the hospital.
