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“I'm talking with people involved every single day. I believe it's going happen,” he told reporters at the White House.
When asked if he had a message for the families of the hostages, Mr Biden said: “Hang in there. We're coming.”
He did not go into additional detail.
Israel believes that the more than 200 people who remain unaccounted for following the October 7 attacks are currently being held hostage by Hamas. The White House has said that nine US citizens remain missing.
National Security Adviser Jake Sullivan has said that the White House's knowledge of the condition of the hostages is limited, but that a “substantial number” could be part of a release.
Mr Biden has also been involved in discussions with Israel and Qatar on the potential release of hostages.
“We’re not there yet. But we are continuing to try to make progress on that day by day, hour by hour,” Mr Sullivan said on Monday.
The hostage situation has evolved into a key sticking point in calls for a ceasefire, which Israel and the US have thus far rejected. Prime Minister Benjamin Netanyahu said there would be no ceasefire until hostages are released from Gaza.
Hamas has said that it could agree to a Qatar-mediated deal that would lead to the release of Israeli hostages in exchange for Palestinians being held in Israeli prisons. A ceasefire would also have to be included as part of that deal, Hamas said.
Two American citizens were among a group of hostages who were earlier released by Hamas.
In a phone call with Qatar's Emir Sheikh Tamim on Sunday, Mr Biden spoke on “the urgent ongoing efforts to secure additional releases”.
The two leaders also agreed all hostages “must be released without further delay”, the White House said in a readout of the call.