The development came as the U.N. Secretary-General Ban Ki-moon pressed Israel on a cease-fire, and Gulf leaders gathered in Saudi Arabia to discuss the conflict.
Meanwhile, Israeli troops pushed deeper into densely populated Gaza City on the 20th day of the offensive to rout out Hamas militants. Israeli tanks shelled the crowded downtown, sending terrified residents fleeing for cover.
An Israeli airstrike Thursday evening killed prominent Hamas figure Said Siam, and witnesses and U.N. officials said Israeli shells struck the United Nations headquarters building that serves as a shelter for hundreds of people, setting it ablaze.
The Israeli push ratcheted up pressure on Hamas to accept a proposed cease-fire. Egypt's proposal has centered on a temporary 10-day halt in fighting that would leave Israeli troops in place in Gaza while security arrangements are negotiated for border crossings to prevent weapons smuggling. Once that is done, Israeli troops would withdraw and the borders would be opened.
In the past two days, some Hamas officials have said the Egyptian proposal could be sealed soon. But on Thursday, the movement's top political leader, Khaled Mashaal, insisted its conditions for a cease-fire remained the same — including a demand for an immediate Israeli withdrawal from Gaza and opening of the crossings.
"These are our demands and we don't accept any political movement that does not accept them," Mashaal said in a televised address from his headquarters in the Syrian capital, Damascus.
"Now we are at the critical moments. There must be a cease-fire but it must meet our conditions," he said.
The Israeli envoy, Amos Gilad, spent four hours in Cairo in talks with Egyptian officials, presenting Israel's "parameters of the end game," said Israeli government spokesman Mark Regev. Gilad did not meet Hamas envoys who are also in town.
After top-level consultations called by Israeli Prime Minister Ehud Olmert when Gilad returned to Jerusalem, it was decided to send him back to Cairo on Friday for further talks, Olmert's office said.
Israel wants a total end to Hamas' rocket launches into Israel and an arms embargo on Gaza's militant rulers, Regev said.
"There is momentum in these discussions," Regev told AP Television News. "We are hopeful that a deal will be based on a total cessation of Hamas fire into Israel and an arms embargo to prevent Hamas from rearming is close and attainable."
Gaza-based Hamas official Ghazi Hamad said the deeper incursion reflected an attempt to pressure his group.
"Israel when it feels that there's a political solution it begins hitting harder," he said. "Amos Gilad came to Cairo and knew there was an agreement between Egypt and Hamas, and it wasn't far from Israel's demands. But they want to increase their military attacks to try impose their conditions."
Hamad said his group has offered amendments to Egypt's original peace proposal, and he expected the Egyptians will convey them to the Israelis. "Consultations are continuing," he said.
Before Mashaal's comments, Salah Bardawil, a Hamas negotiator in Cairo, said his group demands Israeli troops withdraw within five days of the start of a cease-fire and seeks Turkish or European monitors to ensure that border crossings remain open.
The comments appeared to reflect signs of cracks within Hamas, between its Gaza-based officials like Bardawil and Hamad, and the Damascus-based leadership-in-exile.
In Tehran, Iranian President Mahmoud Ahmadinejad said the fighting in Gaza has been "a great lesson for all," saying it shows "the absolute defeat and desperation of this (Israeli) regime."
He says that "even for the supporters of the occupying regime and its leaders, it has become clear that the continuation of the Zionist regime's life in the region is not feasible."
Israel says it launched the offensive Dec. 27 to stop rocket fire against southern Israeli towns by Hamas, which has controlled the Gaza Strip since 2007.
Meanwhile, an emergency summit of the six Gulf Cooperation Council countries, called by Saudi Arabia to discuss Gaza, was held in Riyadh on Thursday evening.
Arab foreign ministers were to meet Friday in Kuwait and were expected to make aid pledges for Gaza.
Associated Press Writers Patrick Quinn in Jerusalem and Nasser Karimi in Tehran contributed to this report