With an Egyptian official acting as an intermediary, Hamas on Monday gave Israel its conditions for a cease-fire, a general in Egyptian intelligence told CNN.
A special Israeli delegation received the letter in Cairo from intelligence chief Mohamed Shehata, the general said.
"We are optimistic about the negotiations so far," the general said.
No details of the conditions were immediately available, nor was there any immediate confirmation of the letter from Israel.
The movement comes amid mounting international pressure to end the hostilities, including U.N. Secretary-General Ban Ki-moon flying to the region to personally appeal for a cease-fire.
Ban's call for a truce came on the heels of the single deadliest attack -- an Israeli airstrike that killed a family of 10 -- in the conflict that began with militant rocket attacks from Gaza, to which Israel responded with an aerial offensive.
"This must stop," Ban said late Sunday. He called on both sides to cooperate with Egyptian-led effort to broker a cease-fire.
"I am heading to the region to appeal personally for ending the violence and contribute to ongoing efforts to that end," he said.
Ban joins a growing chorus of Western and Arab diplomats calling for end to the crisis that has raised fears of a repeat of Israel's 2008 invasion of Gaza following a similar spate of rocket attacks. At least 1,400 people were killed in that conflict.
In Israel, three people have been killed and 68 wounded in rocket attacks, the Israel Defense Forces says.
At least 92 people have been killed in Gaza since Israel began its aerial campaign, Palestinian emergency services said Monday. About 750 people in Gaza have been injured, the group said.
Neighbors and family members used a ditch digger, shovels and, in some cases, their hands to dig through the debris of a two-story house blown apart by an Israeli airstrike.
Ten members of one family were buried in the broken concrete and mangled metal, relatives told CNN.
A ditch digger was used to lift a giant slab of concrete. Underneath, the bodies of two small children were discovered.
Nearby, men dug through concrete blocks to find a missing woman.
"She's my uncle's wife," a young man shouted as he tried to get to a debris pile where the woman was believed buried. "She lived here."
A short time later, the body of the elderly woman was uncovered.
On Twitter, the al Qassam Brigades -- Hamas' military arm -- called it a "massacre committed by Israeli occupation."
The Israeli airstrike was targeting Yehya Bayaa, "a senior Hamas member," said Lt. Col. Avital Leibovich, the IDF's chief spokeswoman. The IDF alleges Bayaa is one of the leaders of a Hamas rocket-launching unit.
"When I say a senior Hamas member, I mean members that have Israeli blood on their hands -- members of Hamas that planned either the abduction of soldiers or are very much involved in targeting Israelis," Leibovich said.
The house was Bayaa's home and suspected command center, according to Leibovitch. She said the Israeli military was examining video of the strike to look for signs of secondary explosions, an indication that there were explosives inside. Initially, the IDF reported it killed Bayaa in the attack. But late Sunday, Leibovich said she did not know for sure whether Bayaa had been killed.
Rockets flew overhead as mourners gathered Monday at the al-Isra mosque for the funeral of some of the family members killed, CNN's Ben Wedeman reported. A short time later, the sound of the firing of more rockets could be heard.
Hundreds turned out for the funeral, where some chanted, "revenge, revenge."
Q & A: What is Hamas?
Also on Monday, militants fired six rockets toward Eshkol in southern Israel, with one hitting a school that has been shuttered since the conflict began, the IDF said. Three more rockets fired at Ashkelon -- which has been repeatedly targeted by militants in recent days -- were intercepted by Israel's missile defense system, it said.
Airstrikes Monday in Gaza targeted a stadium where the IDF alleges Hamas militants were launching rockets.
Overnight, a fairly intense air campaign was carried out with a number of airstrikes targeting primarily government buildings.
In the daylight, Gaza City looked something like a war zone with flattened buildings, emptied streets and shuttered store fronts.
An Israeli special envoy was in Egypt for cease-fire talks Monday, the Egyptian government said, and a steady stream of Arab League, U.N. and European diplomats were arriving in the region to promote a cease-fire.
Hamas is putting conditions on any truce offer. In addition to an end the airstrikes, senior Palestinian negotiator Nabil Shaath said Hamas is demanding the end to Israel's long blockade of Gaza.
The territory has been under a crippling economic embargo since Hamas won control of the territory from the Fatah-led Palestinian Authority,