US President Donald Trump told reporters at the White House: "We have a really good chance of making a deal, and it'll be a lasting deal."
Palestinian and Egyptian officials have told the BBC that the sessions are focused on "creating the field conditions" for a possible exchange that would see the release of all Israeli hostages in return for a number of Palestinian prisoners, reports BBC.
Hamas has said it agrees to the peace plan proposals in part, but has not responded to several key demands - including its disarmament and not having any future role in the governance of Gaza. Israel's prime minister said on Saturday that he hoped to announce the release of hostages "in the coming days".
Indirect talks aimed at reaching a final agreement on a US peace plan to end the war in Gaza are set to continue on Tuesday in the Egyptian city of Sharm El-Sheikh.
UK Prime Minister Sir Keir Starmer reiterated his support for the plan in his statement commemorating the anniversary, saying: "We welcome the US initiative towards peace in the Middle East, and this government will do everything in our power to bring about the day where every child of Israel can live peacefully, alongside their Palestinian neighbours, in safety and security."
A Palestinian official close to the negotiations told the Reuters news agency that the first session ended late Monday evening, and more talks were due to take place on Tuesday.
The statement made no mention of one of the key demands of the plan – that Hamas agree to its disarmament and to playing no further role in the governance of Gaza.
It added that the part of the proposals dealing with the future of Gaza and the rights of Palestinian people was still being discussed "within a national framework", of which it said Hamas will be a part.
Many Palestinians described Hamas' response to the peace plan as unexpected, after days of indications that the group was preparing to reject or at least heavily condition its acceptance of Trump's peace plan proposal.
European and Middle Eastern leaders have welcomed the proposal. The Palestinian Authority (PA), which governs parts of the Israeli-occupied West Bank, has called the US president's efforts "sincere and determined".
Iran - which has been one of Hamas's main sponsors for many years - has also now signalled its support for Trump's Gaza peace plan.
Mahmoud Basal, spokesman for Gaza's Hamas-run civil defence, told the BBC that "no aid trucks have been allowed into Gaza City since the offensive began four weeks ago".
"There are still bodies we cannot retrieve from areas under Israeli control," he said.
Hundreds of thousands of Gaza City residents have been forced to flee after the Israeli military ordered evacuations to a designated "humanitarian area" in the south, but hundreds of thousands more are believed to have remained.
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