He said indirect negotiations were under way about the return of about 100 hostages and that, while it was still too early to be sure, prospects had improved.
"We can be more optimistic than before but we are not there yet. I hope we will be there," Saar told a press conference in Jerusalem, reiterating Israel's position that the hostages still held in Gaza must be returned before Israel agrees to an end to the fighting.
"There will not be a ceasefire in Gaza without a hostage deal," he said.
On Monday night, Prime Minister Benjamin Netanyahu told a press conference that Hamas' increasing isolation following the collapse of Syrian President Bashar al-Assad's rule potentially opened the door to a hostage deal but it was too early to say efforts would succeed.
A Palestinian official with knowledge of the mediation effort said Hamas had asked other factions in Gaza to start listing the names of Israeli and foreign hostages in their custody, whether dead or alive.
The official gave no further details of the mediation effort but said the mediators, backed by the United States, had stepped up contacts with Israel and Hamas.
Hamas officials declined immediate comment.
An official of the group allied with Hamas expressed hope that talks could lead to a deal.
Hamas gunmen took over 250 hostages back to Gaza after their attack on Israel on Oct. 7, 2023, which killed some 1,200 people, according to Israeli tallies. Over 44,700 Palestinians have been killed in the Israeli offensive on Gaza that followed, Gaza health authorities say.
Some hostage families voiced cautious optimism after meeting Netanyahu on Sunday. Netanyahu told them the time had come for a hostage deal, the Hostages and Missing Families Forum said.
Each side accuses the other of standing in the way of a deal but Saar said Hamas' previous position "might have changed during recent times".
"So if both parties are interested in an agreement, there is a better chance it will be achieved," he said.
Israeli strikes across Gaza continued overnight and on Monday, medics said. One strike killed at least four people near Jabalia camp on the northern edge of the enclave, they said.
In Rafah, near the southern Gaza border with Egypt, health officials said rescue teams have recovered at least 11 bodies of Palestinians killed in Israeli airstrikes overnight and on Monday.
Residents of Al-Maghazi camp in the central Gaza Strip said some Israeli tanks pushed into the eastern area of the camp early on Monday, forcing some residents to flee their homes.
Later on Monday, medics said an Israeli strike in Al-Maghazi killed four children aged between 4 and 13, while an airstrike in Beit Lahiya in the northern Gaza Strip killed six people.
The Israeli military has not commented on Monday's strikes on Gaza.
