GAZA CITY, Gaza Strip — Terrified residents ran for cover Tuesday in a densely populated neighborhood of Gaza City as Israeli troops backed by tanks thrust deeper into the city and sought Hamas fighters in alleys and cellars.

On the diplomatic front, Egyptian mediators pushed Hamas to accept a truce proposal and, in a hopeful sign, Israel sent its lead negotiator to Cairo for "decisive" talks on a cease-fire. U.N. Secretary-General Ban Ki-moon also headed for the region to join diplomatic efforts.

Israeli military officials say that depending on what happens with what they described as "decisive" talks in Cairo, Israel will move closer to a cease-fire or widen its offensive. They spoke on condition of anonymity because they were discussing sensitive policy matters.

Asked if Israel's war aims had been achieved, Israeli Defense Minister Ehud Barak said: "Most of them, probably not all of them."

Israeli troops now have the coastal city of 400,000 virtually surrounded as part of an offensive launched Dec. 27 to end years of Palestinian rocket attacks on its southern towns.

Early Wednesday an Israeli warplane fired a missile at the former Gaza city hall, used as a court building in recent years, witnesses said. The 1910 structure was destroyed and many stores in the market around it were badly damaged, they said.

Palestinian medical officials reported at least 42 deaths from the conflict on Tuesday throughout Gaza.

The Israeli military said three soldiers were wounded, including an officer who was searching a northern Gaza house when a bomb exploded.

Palestinian hospital officials say more than 940 Palestinians, half of them civilians, have been killed in the fighting. A total of 13 Israelis, 10 of them soldiers, have died.

Palestinian rocket fire has dropped significantly since the offensive was launched. Some 15 rockets and mortar shells were fired toward Israel Tuesday, causing no injuries, the army said.

Fireballs and smoke plumes from Israeli bombing have become a common sight in the territory of 1.4 million people, who are effectively trapped because of blockaded border crossings. Recent fighting has focused on Gaza City, where Israeli soldiers could be increasingly exposed to the treacherous conditions of urban warfare.

The operation in Tel Hawwa neighborhood, one mile (1.5 kilometers) southeast of downtown, matched fast-paced forays into other areas designed to avoid Israeli casualties. Residents said troops entered overnight, reconnoitered the area, and then pulled back to more secure positions.