Egyptian officials mediating talks for a durable truce in Gaza met a Palestinian delegation Wednesday to relay terms laid down by an Israeli team, Palestinian officials said.
The Egyptian intelligence mediators had met with the Israeli delegation in Cairo during the night, the officials said.
Wednesday's shuttle diplomacy came as a fragile ceasefire in the Gaza Strip after a month of fighting between Hamas and Israeli forces appeared to be holding for a second day.
Palestinian officials in Cairo said they would respond to the Israeli terms through the Egyptian officials, although a senior Hamas official has already rejected an Israeli demand that militants in Gaza disarm.
Meanwhile, international Middle East peace envoy Tony Blair and UN Middle East peace process coordinator Robert Serry held separate talks with Egyptian officials on Wednesday.
The Egyptian-mediated 72-hour ceasefire that went into effect on Tuesday has brought relief to both sides after fighting that erupted on July 8 killed 1,875 Palestinians and 67 people on the Israeli side.
The talks in Cairo are aimed at securing a durable ceasefire after the three-day window closes.
Experts say the underlying problems in Gaza, a small coastal enclave flanked by Israel and Egypt that has undergone three conflicts with Israel since 2008, would also have to be addressed.
Signup to our newsletter and follow us on Facebook and Twitter!
US Secretary of State John Kerry, in a BBC interview, called for a sustained ceasefire but stressed that the crucial wider issues would need to be tackled.
The Palestinians demand an end to the eight-year Israeli blockade of Gaza and the release of Palestinian prisoners.
"The six Israeli crossings should be dealt with and opened to provide basic humanitarian needs and to rebuild Gaza," said Egypt's Foreign Minister Sameh Shoukri at a press conference.
"This is one of the main issues that will be tackled in the negotiations... to open (the crossings) after years of suffering by the Palestinian people."
The Palestinian delegation is also demanding an airport be allowed to operate in Gaza, senior Hamas official and delegation member Ezzat al-Rishq said.
Israel, for its part, wants Hamas, which won Palestinian elections in 2006 although designated a terrorist group by the United States and Israel, to disarm its militia.
Rishq said Hamas would not even consider laying down their weapons, which include an arsenal of unguided rockets and anti-tank missiles.
"Whoever tries to take our weapons, we will take his life," he said on Twitter.