The Prime Minister of Qatar, the nation hosting the talks, said on Sunday only “minor details” remained to be overcome.
Finer said even though a deal was not yet in place, “significant progress” had been made.
“I would say that talks have clearly reached a very sensitive stage,” he told Meet the Press on NBC.
“We’re following this minute by minute, hour by hour, and have been over a number of weeks.”
He added: “Some of the issues, whether it was disagreements, have now been either narrowed or an understanding has been reached, but it is not complete, it is not everything.”
When asked if the number of hostages included in a potential deal amounted to “12, more than two dozen, dozens”, he replied: “We’re talking about considerably more than 12.”
He said a temporary ceasefire would allow more humanitarian assistance to reach Gaza.
Finer said the White House believed “a significant number, most likely the majority and even the vast majority, of hostages are alive”.
Herzog told ABC’s This Week that Israel had ruled out a ceasefire, but said there could be a pause in the fighting “so we can get the hostages out”.
Netanyahu’s government has come under intense domestic pressure from some quarters to prioritise freeing the hostages over defeating Hamas, with thousands taking part in a march to Jerusalem to demonstrate outside his office on Saturday.
Sheikh Mohammed Bin Abdulrahman al-Thani, a Qatari diplomat, said on Sunday that the main challenges were “practical and logistical”, but he was confident “we are close enough to reach a deal”.
“The challenges that remain in the negotiations are very minor compared to the bigger challenges,” he said.
“The deal is going through ups and downs from time to time throughout the last few weeks.”
On Sunday, Beijing announced foreign policy officials from the Palestinian Authority and Muslim-majority countries would convene in China for two days to promote a “de-escalation of the current Palestinian-Israeli conflict”.
Officials, including the foreign ministers of the Palestinian Authority in the West Bank, Saudi Arabia, Jordan, Egypt and Indonesia, as well as the secretary-general of the Organisation of Islamic Co-operation, are expected to meet on Monday and Tuesday.
“During the visit, China will have in-depth communication and co-ordination with the joint delegation of foreign ministers of Arab and Islamic countries to promote a de-escalation of the current Palestinian-Israeli conflict, protecting civilians, and justly resolving the Palestinian issue,” Chinese Foreign Ministry spokeswoman Mao Ning said.
The outline for the deal was said to have been put together during weeks of talks in Qatar. The pause is also intended to allow a significant amount of humanitarian aid in from Egypt.
Netanyahu was due to meet the families of hostages with his war cabinet on Monday.
Joe Biden said a ceasefire would not bring peace.
In an opinion article in the Washington Post on Saturday, the US president said Hamas would merely exploit it to “rebuild their stockpile of rockets, reposition fighters and restart the killing by attacking innocents again”.
“As long as Hamas clings to its ideology of destruction, a ceasefire is not peace,” he wrote.
Biden said Hamas hid “among Palestinian civilians”, used “children and innocents as human shields” and positioned “terrorist tunnels beneath hospitals, schools, mosques and residential buildings”.
“If Hamas cared at all for Palestinian lives, it would release all the hostages, give up arms, and surrender the leaders and those responsible for October 7,” he said.
He proposed a “revitalised” Palestinian Authority should govern Gaza and the West Bank after the war as one unit until a two-state solution could be definitively agreed upon.
When asked about the US President’s proposal, Netanyahu said in Tel Aviv that the Palestinian Authority in its present form was not capable of being responsible for Gaza.
Israel has not disclosed a strategy for the territory after the war.
