In a step that could bring a deal closer, Washington promised security guarantees addressing Israeli demands in Egyptian-brokered truce negotiations.
Saeed Seyyam, who as interior minister in Hamas's government oversaw 13,000 Hamas police and security men, was killed in an air strike in Jabalya refugee camp, Hamas said.
Palestinian sources said he had been in a house rented by his brother, who also killed along with Seyyam's son.
"The blood of Seyyam will be a curse on the Zionist entity," Hamas official Mohammed Nazzal told Al Jazeera television.
At least 15 Palestinians were killed in Israeli attacks in Gaza city, medical officials said.
Hamas official Ayman Taha called the street warfare, in which Israeli fire struck a UN compound, a hospital and a media building, an attempt to force the group to accept Israel's terms for a truce. A hospital was also hit in the fighting.
In Cairo, an Israeli defence official held talks with Egyptian mediators who have been negotiating with Hamas officials on a ceasefire. He was to report back to Israeli leaders later in the day.
Israeli Prime Minister Ehud Olmert's office said U.S. Secretary of State Condoleezza Rice told him by telephone that Washington would sign an agreement on measures to stop Hamas from rearming after a ceasefire, Olmert's office said.
"The secretary of state noted that the United States would be prepared to assist in solving the issue of smuggling and to sign a memorandum of understanding with Israel on the subject," Olmert's bureau said in a statement.
Israel has said that a ceasefire must ensure that Hamas can no longer smuggle in weapons through tunnels under the Gaza-Egypt border, as well as end the group's rocket attacks on its southern towns.
Hamas wants Israel, which launched its Gaza offensive on Dec. 27, to withdraw its troops and lift a long-standing blockade of the Gaza Strip.
The United Nations Relief and Works Agency (UNRWA) said its compound, where up to 700 Palestinians were sheltering, was struck twice by Israeli fire and three staff members were injured. Thick smoke rose from its food and fuel depot.
UN Secretary-General Ban ki-Moon called it an "outrage". Meeting Ban later, Olmert apologised but said the shelling was prompted by fire from Palestinian gunmen at the compound.
"It is absolutely true that we were attacked from that place," Olmert said in broadcast remarks.
In Geneva, a spokesman for the International Federation of the Red Cross and Red Crescent Societies said al-Quds hospital was hit by either Israeli shelling or air strikes.
"We can't be certain in the heat of the conflict whether it was deliberate targeting," the spokesman, Paul Conneally, said.
No one was reported hurt at the hospital, where its administrative offices were set ablaze and dozens of patients moved in panic to the ground floor, seeking safety.
A rocket hit the downtown Al-Shurouq Tower, where the Reuters bureau and other media offices are located. Gulf-based Abu Dhabi Television said it believed its two journalists were targeted by an Israeli aircraft as they filmed from a 14th-storey office.
The Israeli military had no comment throughout the day on the attacks on the hospital and media offices.
About 25 rockets fired from the Gaza Strip hit southern Israel on Thursday, wounding six people, police said.
The Palestinian death toll from the air-and-ground offensive has risen to at least 1,095 and there were more than 5,000 wounded, according to the Hamas-run Health Ministry in Gaza. A Palestinian rights group said at least 670 of the dead were civilians.
Thirteen Israelis have been killed - 10 soldiers and three civilians hit by Hamas rocket fire since Israel launched on Dec. 27 a military campaign it said was aimed at ending such salvoes.
Dozens of terrified residents of neighbourhoods near the Gaza city centre were seen fleeing on foot. Thousands more huddled in their homes as explosions tore through rubble-strewn streets clouded by smoke.
"It is a catastrophe," one woman said, walking quickly away from the area and carrying a child in her arms as two other children ran behind her to keep up.
"We took our money and passports. We have to carry some identification with us in case we get killed," she said. "Hamas can claim victory if it wants but we just need this bloodshed to end."
A senior Western diplomat said Israel appeared to be trying to make last-minute gains on the ground before a truce could be imposed.
"It's a classic Israeli strategy," the diplomat said.
Diplomats said Egypt's proposals centred on a phased-in ceasefire, starting with a lull to let in aid, followed by an Israeli pullout and border crossing openings.
In talks in Washington and Cairo, Israeli officials have said they wanted security guarantees to fall under an American "umbrella" and include Egyptian acceptance of U.S. and European advisers and technology to help combat smuggling through the border tunnels, diplomats said.
In addition to bolstering security along the so-called Philadelphi corridor that separates Gaza from Egypt, Israel has demanded an international maritime monitoring programme to track and halt vessels carrying rockets to Hamas.
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