"I believe so", the president told reporters on Monday when asked at the White House about a possible agreement closing.
Qatari mediators had been seeking a deal between Israel and Hamas to exchange 50 hostages in return for a three-day ceasefire that would help boost emergency aid shipments to Gaza civilians, according to reports last week.
About 240 hostages were taken during Hamas's deadly rampage into southern Israel on October 7, which prompted Israel to invade Gaza with the aim of wiping out the Islamist militant group.
About 1,200 people, mostly civilians, were killed in the Hamas assault, according to Israel, making it the deadliest day in Israel's 75-year-old history.
During a press conference in London on Monday at the Israeli Embassy, an Irish father of an nine-year-old girl taken as a hostage pleaded for her release as he spoke of his “nightmare” and how his daughter would be thinking “why has my daddy not come to save me.”
Emily Hand was originally feared dead after the Hamas assault on Kibbutz Be’eri in Israel on October 7. However the Irish-Israeli schoolgirl was one of the 30 children taken.
Also on Monday Israeli forces released security camera video showing what they said were two foreign hostages, one Thai and one Nepalese, who were captured by Hamas in the October 7 attack and taken to Gaza's largest Al Shifa hospital. Hamas said its fighters brought them in for medical care.
The army also said an investigation had determined that Israeli army Corporal Noa Marciano, 19, another captive whose body was recovered in Gaza last Friday, had been wounded in an Israeli strike on November 9 that killed her captor, but was then killed by a Hamas militant in Gaza's Al Shifa hospital.
The Israeli military has previously released images of several guns it said were found inside an MRI lab and said that the bodies of two hostages were found near the complex.
Hamas and hospital staff have denied the allegations of a command post under Shifa.
Senior Hamas official Osama Hamdan dismissed the latest announcement, saying "the Israelis said there was a command and control center, which means that the matter is greater than just a tunnel.
Gaza's Hamas-run government said at least 13,000 Palestinians have been killed since the Hamas attack on October 7, including at least 5,500 children, by Israeli bombardment.
Despite ongoing fighting, Israel's ambassador to the United States, Michael Herzog, told ABC News on Sunday that Israel was hopeful a significant number of hostages could be released by Hamas "in coming days.
It comes as the families of Israeli hostages being held in Gaza pleaded with Israeli lawmakers not to advance legislation that would permit the death penalty for convicted Palestinian militants, including those involved in Hamas' October 7 assault.
The families, who represent some of the 240 people abducted by Hamas and other militants that day, told far-right National Security Minister Itamar Ben-Gvir that the legislation risked angering Hamas and putting the lives of their relatives at risk.
"Not now, when the lives of our loved ones are on the line, when the sword is on their necks," Gil Dikman, whose cousin is believed to be held hostage in Gaza, said Monday during an emotional hearing at the Knesset, Israel's parliament.
The National Security Committee hearing devolved into a screaming match between nationalist lawmakers and hostage families.
After the hearing, Ben-Gvir wrote on X, formerly Twitter, that the legislation is an important tool for increasing pressure on Hamas.
"In the Middle East, you don't blink, you hit your enemy with every tool and bring them to their knees," he said.
Ophir Katz, the head of the government coalition and a member of Prime Minister Benjamin Netanyahu's Likud party, told reporters that the amendment to the penal code would not come up for a vote until it was vetted by senior Cabinet members and Mr Netanyahu.