The day after Israeli and US officials refuted a Washington Post claim that an agreement had been reached, fighting erupted on Sunday, with Hamas fighters confronting Israeli forces attempting to drive 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," Finer said on NBC's "Meet the Press" show.
"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 said.
"Nothing is agreed until everything is agreed," Finer said. Such delicate discussions can go apart at the last minute."
On Oct. 7, Hamas grabbed around 240 captives during its violent cross-border incursion into Israeli settlements, prompting israel to lay siege to Gaza and invade the Palestinian enclave in order to eliminate its dominant Islamist party.
"We're talking about considerably more than 12 (hostages)," Finer told NBC.
"This could and would likely include an extended period of a pause in the fighting, a multiple-day period," he said. "We believe it will allow us to get more humanitarian aid into Gaza." Under all conditions, that is a top priority."
Finer also stated that israel should not begin combat operations against Hamas in Gaza's south until military planners had considered the safety of fleeing Palestinian people.
"In the event that israel is likely to embark on combat operations, including in the south, we believe... that they have the right to do that," Finer said in a separate interview with CBS' Face the Nation.
"We think that their operations should not go forward until those people, those additional civilians, have been accounted for in their military planning," he went on to say.
Israel's bombardment has left large areas of the north to ruins, while over two-thirds of Gaza's 2.3 million people have been displaced to the south. Gaza's health Ministry has raised its death toll from Israeli bombardment to 12,300, including 5,000 children.