In Cairo, Palestinian Authority President Mahmoud Abbas and Arab League Secretary-General Nabil Al-Arabi held talks Thursday aimed at reaching a cease-fire agreement.
An Israeli delegation also attended, leaving after several hours, the state-run al-Ahram news agency reported.
"I expect that we will reach an agreement very soon; the efforts of a cease-fire is to stop the bloodshed, killing and destruction in Gaza," said Nabil Shaath, an Abbas adviser and member of the central committee of the Palestine Liberation Organization.
He said negotiators were focusing on stopping bloodshed above all else. He said they would later discuss Hamas demands, including opening Gaza border crossings and freeing prisoners whose exit from jail was negotiated in exchange for the release of Israeli soldier Gilad Shalit.
"These are all legitimate demands by Hamas, but the priority is for an immediate cease-fire," Shaath said.
Hamas leaders had rejected an earlier Egyptian cease-fire proposal, saying they had not been consulted on the deal and complaining that it did not address their broader demands.
Hamas officials had said Wednesday they would not participate in the Cairo talks, but PLO official Saeb Erakat -- who was in the Egyptian capital with Abbas -- said the Palestinian Authority leader had met with representatives of Hamas and Islamic Jihad.
Erakat said negotiators are trying to extend the U.N.-sponsored temporary humanitarian cease-fire on a "rolling basis."
"While there is no plan at this point for a comprehensive cease-fire agreement ... we are trying to extend the current one by another six or 10 hours, or even several days if possible," he said.
Egypt is playing a large role in the talks despite its distrust of Hamas.
Like Israel, Egypt considers Hamas a terror organization because of the group's roots in the Muslim Brotherhood, which Egypt's military-led government banned after the country's 2013 coup.
During Thursday's temporary cease-fire, Gaza banks opened for the first time in 10 days. Residents poured into the streets.
Red Cross officials visited hospitals and damaged houses to assess medical needs, and worked with local officials to fix water pipelines. Some work was also done to repair power lines, the United Nations said.
At least 10 to 15 trucks entered Gaza through the Karem Shalom border crossing once it opened at 1 p.m. (6 a.m. ET), according to Ra'ed Fatooh, the Palestinian official in charge of the crossing in Gaza.
Goods were limited to medical supplies and basic foodstuffs such as rice, sugar, oil, canned food, flour and other basic goods, he said.
"The trucks are being subjected to strict and difficult search by the Israeli security before entering the crossing," he said.
The search, he said, illustrated one of the chief complaints of Gaza residents -- fluctuating border controls that often stymie the flow of goods into the territory.
"We want the crossing to open in a normal fashion to go back to how it was before 2007 and to bring the required goods and products for Gaza for the people and residents to live in dignity as the rest of the world," Fatooh said.
Deaths on Gaza beach