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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 on Monday, while the U.S. joined a stream of countries pushing for a cease-fire.

At least 14 Palestinian children were killed, raising the known death toll from a new ground invasion to more than 80. The vast majority of confirmed deaths have been civilians, fueling international outrage. Gaza’s biggest hospital said it was overwhelmed.

As the bruising campaign against Gaza’s Hamas rulers entered its 10th day, the Islamic militant group continued to pummel southern Israel with more than two dozen rockets on Monday and promised to wait for Israeli soldiers “in every street and every alleyway.”

Israeli Defense Minister Ehud Barak said the offensive would go on until Israel achieved “peace and tranquility” for residents of southern Israel.

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. On Monday, Israeli forces pounded houses — one of them belonging to a leading Hamas member who was not there at the time — a pair of mosques and smuggling tunnels.

Israel has attacked several mosques during the campaign, saying they were used to store weapons.

The Israeli army said “dozens” of militants have been killed or wounded, but Hamas has not released casualty figures.

Gaza health officials reported some 540 Palestinian dead and nearly 2,000 wounded since Israel embarked upon the campaign on Dec. 27, including at least 200 civilians.

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.

In Washington, the State Department said the U.S. was pressing for a cease-fire 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.

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.

While blaming Hamas for causing Palestinian suffering with rocket fire that led to the Israeli offensive, 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 cease-fire.

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

“The EU insists on a cease-fire 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 cease-fire “must be concluded by the involved parties,” he added.

Livni said the operation was designed to change the rules of Israel’s struggle against Hamas after seven years of rocket fire at Israeli towns. From now on, she said, “When Israel is targeted, Israel is going to retaliate.”

Israel’s operation angered many across the Arab world and has drawn criticism from countries like Turkey, Egypt and Jordan, which have ties with Israel and have been involved intimately in Mideast peacemaking.

The Palestinian foreign minister, Riad Malki, who works within the rival Fatah administration from the West Bank, asked the U.N. Security Council to quickly adopt a resolution calling for an immediate end to Israeli attacks in Gaza and a permanent cease-fire including border monitors and an international force to protect civilians.

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

Gaza’s biggest hospital, Shifa, has been 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 21 who died in various attacks across Gaza on Monday.

Fourteen 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, and a mother and her four children were killed in another strike. 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, a different Gaza health official 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 cell phones, a neighbor said, quoting a relative in one of the buildings who called 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.

Maj. Avital Leibovich, an Israeli military spokeswoman, said Hamas was to blame for civilian casualties because it operates in densely populated areas.

“If Hamas chose cynically to use those civilians as human shields, then Hamas should be accountable,” she said. “Civilians will probably continue to get killed, unfortunately, because Hamas put them in the first lines of fire.”

Black smoke from tank shells and wind-swept dust billowed in the air over Gaza City, home to 400,000 people, where the streets were almost empty. Two children crossing a street near a Hamas security compound didn’t bother to look right and left for cars but gazed up at the sky, apparently looking for attack aircraft.

Unmanned Israeli planes and Apache helicopters circled overhead.

Hamas leaders went into hiding before the Israeli military strike began and only on rare occasions have addressed the Gaza residents in broadcasts from their hideouts.

On Monday, Hamas leader Mahmoud Zahar exhorted Palestinians to fight the Israeli forces and target Israeli civilians.

“The Zionists have legitimized the killing of their children by killing our children. They have legitimized the killing of their people all over the world by killing our people,” Zahar said in a grainy video broadcast on Hamas TV.

A spokesman for Hamas’ military wing, identified by a nom de guerre, Abu Obeida, warned the Israelis that militants “wait for you in every street and every alleyway.”

The ground clashes took place in open areas militants use to launch rockets and mortars at nearby Israeli communities, but did not advance into urban areas where casualties are liable to swell.

The Israeli military said aircraft carried out 30 sorties overnight, including strikes against a mosque in Jebaliya that contained a large store of weapons and an underground arms bunker in the Gaza City area that touched off secondary explosions and collapsed underground smuggling tunnels.

The violence has deepened the suffering in impoverished Gaza, home to 1.4 million people. The military said Monday that 80 truckloads of humanitarian aid and critical fuel supplies would be let in.

Israel’s ground operation is the second phase in an offensive that began as a weeklong aerial onslaught aimed at halting Hamas rocket fire that now threatens major cities and one-eighth of Israel’s population of 7 million people.

Five Israelis have been killed since the offensive began, including a soldier who died in the ground operation. Heavy Israeli casualties could undermine what has so far been overwhelming public support for the operation.