US deputy national security adviser Jon Finer said an agreement to free “considerably more than 12" hostages would also likely include an extended pause in the fighting and allow for the distribution of humanitarian assistance in Gaza.
Fighting continued to rage on Sunday, with Hamas battling Israeli forces trying to push into Gaza's largest refugee camp.
“What I can say at this point is that some of the outstanding areas of disagreement, in a very complicated, very sensitive negotiation, have been narrowed," Mr Finer told NBC.
“I believe we are closer than we have been in quite some time, maybe closer than we have been since the beginning of this process, to getting this deal done," he added.
Israeli Ambassador to the United States Michael Herzog also said in an interview on ABC that Israel is hopeful a significant number of hostages could be released by Hamas “in coming days."
But Mr Finer cautioned: "Nothing is agreed until everything is agreed. Sensitive negotiations like this can fall apart at the last minute."
Hamas took about 240 hostages during its terror attack on October 7, which prompted Israel to lay siege to Gaza and invade the Palestinian territory to eradicate its ruling Islamist group.
“We're talking about considerably more than 12 [hostages]," Mr Finer told NBC.
“This could and would likely include an extended period of a pause in the fighting, a multiple-day period," he added.
“It would enable us, we believe, to get more humanitarian assistance into Gaza. That's a priority under any circumstances."
Earlier on Sunday, Qatar's prime minister said the main sticking points blocking a deal for some of the hostages’ release were “very minor".
“The challenges facing the agreement are just practical and logistical," Qatari prime minister Sheikh Mohammed Bin Abdulrahman al-Thani said at a joint press conference with European Union foreign policy chief Josep Borrell in Doha.
“The deal is going through ups and downs from time to time throughout the last few weeks.
“But I think that you know I'm now more confident that we are close enough to reach a deal that can bring the people safely back to their home."