White House 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 raged on Sunday, with Hamas militants battling Israeli forces trying to push into Gaza's largest refugee camp, the day after Israeli and U.S. officials denied a Washington Post report that a deal had been reached.
"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," Finer told NBC's "Meet the Press" program.
"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's "This Week" that Israel is hopeful a significant number of hostages could be released by Hamas "in coming days."
But Finer cautioned: "Nothing is agreed until everything is agreed. Sensitive negotiations like this can fall apart at the last minute."