Hamas has said it has handed over 20 hostages to the International Committee for the Red Cross.
Red Cross vehicles were later seen crossing into Egypt.
Qatari and Egyptian mediators earlier said Hamas had agreed to release 13 Israelis and seven foreign nationals on Saturday in exchange for 39 Palestinians imprisoned by Israel.
The announcement came after Hamas delayed the releases for several hours, saying Israel had violated the terms of a truce deal that had set the stage for such swaps.
Saturday is the second day of what was meant to be a four-day truce.
Until Saturday afternoon, it still appeared everything was going according to plan. Aid trucks were entering Gaza, Hamas handed a list of more than a dozen hostages slated for release to mediators Qatar and Egypt, and Israel’s Prison Service prepared a list of dozens of Palestinian prisoners for release.
But by nightfall, as the hostages should have emerged from Gaza, Hamas announced that it was delaying the release over what it said were Israeli truce violations.
The group alleged the aid deliveries permitted by Israel fell short of what was promised, and that not enough of the aid was reaching northern Gaza — the focus of Israel’s ground offensive and the main combat zone.
Hamas also said not enough veteran prisoners were freed in the first swap on Friday.
“This is putting the deal in danger and we have spoken to mediators about that,” Osama Hamdan, a senior Hamas official, said earlier in Beirut.
But Egypt, Qatar and Hamas itself later said obstacles had been overcome, and Hamas issued a statement listing six women and 33 boys and teenagers it said were expected to be released by the Israelis.
Two women, Maysoun Jabali and Israa Jaabis, were imprisoned in 2015 after being convicted of carrying out attacks on Israelis. Jaabis suffered severe burns during the incident.
In Tel Aviv, several thousand people packed a central square called “the square of the hostages,” awaiting news of the second release.
“Don’t forget the others because it’s getting harder, harder and harder. It’s heartbreaking,” said Neri Gershon, a Tel Aviv resident. Some families have accused Prime Minister Benjamin Netanyahu’s government of not doing enough to bring hostages home.
In the Balata refugee camp in the West Bank, the family of 16-year-old Wael Mesheh was getting the house ready for his homecoming as part of the second swap.
“We are going to hug him so tight,” his mother, Hanadi Mesheh, said by phone.
The first group of freed Israelis included nine women and four children aged nine and under. They were taken to Israeli hospitals for observation and were declared to be in good condition.
Hours later, 24 Palestinian women and 15 teenage boys held in Israeli prisons in the occupied West Bank and east Jerusalem were freed. The teenagers had been jailed for minor offences including throwing stones. The women included several convicted of trying to stab Israeli soldiers.
“It’s a happiness tainted with sorrow because our release from prison came at the cost of the lives of martyrs and the innocence of children,” said one released prisoner, Aseel Munir al-Titi.
According to the Palestinian Prisoners’ Club, an advocacy group, Israel is holding 7,200 Palestinians, including about 2,000 arrested since the start of the war.
On Saturday, at least two Palestinians were injured at a tense West Bank checkpoint where Israel was to free prisoners. Israeli security forces fired tear gas and rubber bullets at Palestinians gathered at Beitunia checkpoint. It was not clear how the two were injured.
