The decision comes ahead of President Donald Trump's visit to the Middle East on Tuesday. Hamas said it was also intended to facilitate a deal for the entry of humanitarian aid. Gaza has been under an Israeli blockade for 70 days.
Earlier a senior Hamas official told the BBC that the Palestinian armed group was holding direct negotiations with a US administration official in Qatar.
The Israeli PM's office said it had not committed to any ceasefire but only to a "safe corridor" for Mr Alexander's release.
Benjamin Netanyahu's office said preparations to intensify fighting were still continuing, and the release of Mr Alexander had been made possible because of military pressure on Hamas.
A senior Palestinian official familiar with the negotiations with the US in Qatar told the BBC that Hamas's announcement was intended as a goodwill gesture before Trump's arrival.
He said another meeting between Hamas and the mediators was scheduled for early Monday morning to finalise the process of Mr Alexander's release, which would require a temporary halt to Israeli military activity and a suspension of aerial operations during the handover.
President Trump confirmed Alexander's release in a post on Truth Social, calling it "monumental news" and "a step taken in good faith".
Born in Tel Aviv but raised in New Jersey, 21-year-old Alexander was serving in an elite infantry unit on the border with Gaza when he was captured by Hamas militants during the 7 October attack.
