GAZA CITY, Gaza Strip - Israel seized control of high-rise buildings and attacked houses, mosques and smuggling tunnels as it pressed its offensive against the Gaza Strip's Hamas rulers Monday, even as a stream of European leaders headed to the region to push for a truce.

At least 10 Palestinian children were killed, raising the known death toll from a new ground invasion to more than 80, a Palestinian health official said. The vast majority of confirmed deaths have been civilians, fuelling international outrage. Gaza's biggest hospital said it was overwhelmed, its morgue jammed and its hallways filled with the wounded.

As the bruising Israeli offensive entered its 10th day, Hamas pummelled southern Israel with more than two dozen rockets Monday and promised to wait for Israeli soldiers "in every street and every alleyway."

Israeli Defence Minister Ehud Barak said the offensive would go on until Israel achieved "peace and tranquility" for residents of southern Israel. The rocket attacks were originally sparked by Israel's siege of Gaza.

In Washington, the State Department said the United States was pressing for a ceasefire that would include three main elements, including a halt to rocket attacks.

"We're doing a lot of work on these three elements," said spokesman Sean McCormack said, adding that the goal is to establish a halt to the violence that would meet Secretary of State Condoleezza Rice's standard of being durable and sustainable.

In addition to halting the rocket fire from Gaza, the proposed ceasefire would include an arrangement for reopening crossing points on the border with Israel, McCormack said.

President George W. Bush, however, emphasized "Israel's desire to protect itself."

"The situation now taking place in Gaza was caused by Hamas," he said in the Oval Office.

French President Nicolas Sarkozy, who unsuccessfully proposed a two-day truce before the land invasion began, was due to meet Monday with Israeli Prime Minister Ehud Olmert and Palestinian President Mahmoud Abbas, who lost control of Gaza to Hamas in June 2007.

Sarkozy has condemned Israel's use of ground troops, reflecting general world opinion. Sarkozy and other diplomats making their way to the region are expected to press hard for a ceasefire.

A European Union delegation met with Foreign Minister Tzipi Livni on Monday.

"The EU insists on a ceasefire at the earliest possible moment," said Karel Schwarzenberg, the foreign minister of the Czech Republic, which took over the EU's presidency last week. Rocket attacks on Israel also must stop, Schwarzenberg told a joint news conference with Livni.

The EU brought no truce proposals of its own to the region because the ceasefire "must be concluded by the involved parties," he added.

After a weeklong air offensive, Israeli ground troops invaded Gaza late Saturday. The Israelis have seized a main highway in Gaza, slicing the territory in half. Israeli forces also pounded houses, a pair of mosques and smuggling tunnels Monday as they pressed forward with the offensive.

Israel has attacked several mosques during the campaign, saying they were used to store weapons. One house also attacked belonged to a leading Hamas legislator, who was not inside at the time.

The Israeli army said "dozens" of Hamas fighters have been killed or wounded.

Gaza health officials reported 537 Palestinian dead and nearly 2,000 wounded since Israel embarked upon the campaign Dec. 27. At least 200 civilians were among the dead.

Israel has three main demands: an end to Palestinian attacks, international supervision of any truce and a halt to Hamas rearming.

Hamas demands a cessation of Israeli attacks and the opening of vital Gaza-Israel cargo crossings, Gaza's main lifeline.

Israeli forces seized sparsely populated areas in northern Gaza and by Monday morning were dug in on the edges of Gaza City.

Further movement into the heart of the built-up areas would mean deadly urban warfare, with house-to-house fighting, sniper fire and booby traps in crowded streets and alleyways familiar to Hamas' 20,000 fighters.

Israeli forces have been training in a mock Arab city for more than two years to prepare for urban warfare in Gaza, said the military.

Gaza's biggest hospital, Shifa, was swamped by the bloodshed. Bodies were two to a morgue drawer, the wounded were being treated in hallways because beds were full, and three preschool boys killed in an artillery strike Monday were laid out on a floor.

Since Israel mounted its ground offensive three days ago, most of the dead and wounded arriving at Shifa have been civilians, including 16 who died in various attacks across Gaza on Monday.

Ten of them were children, said health official Dr. Moaiya Hassanain.

Four young siblings were killed in a missile strike on a house east of Gaza City, Hassanain said.

Three other children died in a naval shelling of a Gaza City beach camp, and three toddlers died in an attack on another town outside Gaza City, another Gaza health official said.

In addition, three adults died when a missile struck near a house of mourning in the northern town of Beit Lahiya, and three other adults died in attacks elsewhere, Hassanain said.

Israeli troops seized three six-story buildings on the outskirts of Gaza City, taking up rooftop positions after locking residents in rooms and taking away their cellphones, a neighbour said, quoting a relative in one of the buildings before his phone was taken away.

"The army is there, firing in all directions," said Mohammed Salmai, a 29-year-old truck driver. "All we can do is take clothes to each other to keep ourselves warm and pray to God that if we die, someone will find our bodies under the rubble."

Civilian casualties have spiked since Israel launched the ground offensive. Of the 80 confirmed deaths, at least 70 were civilians, Hassanain said.