GAZA CITY, Gaza Strip — An Israeli missile ripped through a two-story home in a residential area of Gaza City on Sunday, killing at least 11 civilians, including four young children and an 81-year-old woman, in the single deadliest attack of Israel’s offensive against Islamic militants.

A similar scene unfolded elsewhere in the city early Monday, when an airstrike leveled two houses belonging to a single family, killing two children and two adults and injuring 42 people, including children, said Gaza heath official Ashraf al-Kidra. Rescue workers were frantically searching for 12 to 15 members of the Azzam family under the rubble.

While the airstrikes relentlessly targeted militant rocket operations, Israeli gunboats unleashed a steady tattoo of heavy machine gun fire and shells at militant facilities on Gaza’s coastal road.

The bloodshed was likely to raise pressure on Israel to end the fighting, even as it pledged to intensify the offensive by striking the homes of wanted militants. High numbers of civilian casualties in an offensive four years ago led to fierce criticism and condemnation of Israel.

In all, 81 Palestinians, half of them 37 civilians, have been killed in the five-day onslaught and 720 have been wounded. Three Israeli civilians have died from Palestinian rocket fire and dozens have been wounded..

President Barack Obama said he was in touch with players across the region in hopes of halting the fighting, while also warning of the risks of Israel expanding its air assault into a ground war.

“We’re going to have to see what kind of progress we can make in the next 24, 36, 48 hours,” Obama said during a visit in Thailand.

U.N. chief Ban Ki-moon urged the two warring parties to achieve an immediate cease-fire. He said he was heading to the region to appeal personally for an end to the violence, but no date was given in the U.N. statement for his arrival.

On the ground, there were no signs of any letup in the fighting as Israel announced it was widening the offensive to target the military commanders of the ruling Hamas group.

The Israeli military carried out dozens of airstrikes throughout the day, and naval forces bombarded targets along Gaza’s Mediterranean coast. Many of the attacks focused on homes where militant leaders or weapons were believed to be hidden.

Palestinian militants continued to barrage Israel with rockets, firing more than 100 on Sunday, and setting off air raid sirens across the southern part of the country. Some 40 rockets were intercepted by Israel’s U.S.-financed “Iron Dome” rocket-defense system, including two that targeted the metropolis of Tel Aviv. At least 10 Israelis were wounded by shrapnel.

Israel’s decision to step up its attacks in Gaza marked a new and risky phase of the operation, given the likelihood of civilian casualties in the densely populated territory of 1.6 million Palestinians. Israel launched the offensive Wednesday in what it said was an effort to end months of intensifying rocket fire from the Hamas-ruled Gaza Strip.

In the day’s deadliest violence, the Israeli navy fired at a home where it said a top wanted militant was hiding. The missile struck the home of the Daloo family in Gaza City, reducing the structure to rubble.

Frantic rescuers, bolstered by bulldozers, pulled the limp bodies of children from the ruins of the house, including a toddler and a 5-year-old, as survivors and bystanders screamed in grief. Later, the bodies of the children were laid out in the morgue of Gaza City’s Shifa Hospital.

Among the 11 dead were four small children and five women, including an 81-year-old, Gaza health official Ashraf al-Kidra said.

More than a dozen homes of Hamas commanders or families linked to Hamas were struck on Sunday. Though most were empty — their inhabitants having fled to shelter — at least three had families in them. Al-Kidra said 20 of 27 people killed Sunday were civilians, mostly women and children.

Sami Abu Zuhri, a Hamas spokesman, said that “the Israeli people will pay the price” for the killing of civilians.

Israel sought to place the blame on militants, saying they were intentionally operating in places inhabited by civilians. The military has released videos and images of what it says are militants firing rockets from mosques, schools and public buildings.

“Hamas is using the Gaza population as human shields,” said Brig. Gen. Yoav Mordechai, the chief Israeli spokesman. “They are exploiting crowded residential urban areas.”

He acknowledged, however, that it was not clear whether the militant targeted in Sunday’s attack was killed, despite earlier claims of success. “I still don’t know what became of him,” Mordechai told Channel 10 TV.

The prospect of mounting civilian casualties could quickly change the momentum of Israel’s operation. Israel launched the offensive on Wednesday with a lightning airstrike that killed Hamas’ military chief. Since then, it has carried out a blistering campaign of more than 1,200 airstrikes, targeting suspected rocket storage and launching sites.

Israel also struck two high-rise buildings housing media outlets, damaging the top-floor offices of the Hamas TV station, Al Aqsa, and a Lebanese-based broadcaster, Al Quds TV, seen as sympathetic to the Islamists. Six Palestinian journalists were wounded, including one who lost a leg, the Gaza press association said.

Foreign broadcasters, including British, German and Italian TV outlets, also had offices in the buildings.

Lt. Col. Avital Leibovich, an Israeli military spokeswoman, said the strikes targeted Hamas communications equipment on the rooftops. She accused the group of using journalists for cover.

Israeli officials expressed readiness to take the offensive even further with a ground invasion of Gaza. Israel has mobilized thousands of forces and columns of armored vehicles along the border ahead of a possible incursion.

“The Israeli military is prepared to significantly expand the operation,” Israeli Prime Minister Benjamin Netanyahu declared at the start of the weekly Cabinet meeting.

The threats come at an important crossroads — with a fateful choice between further escalation or agreeing to a cease-fire with Hamas. Israel and the West consider Hamas, which seized control of Gaza in 2007, to be a terrorist group.

Obama and British Foreign Secretary William Hague cautioned against a potential Israeli ground invasion of Gaza.

Obama blamed Palestinian militants for starting the round of fighting by raining rockets onto Israel and said the U.S. supported Israel’s right to protect itself. “Israel has every right to expect that it does not have missiles fired into its territory,” Obama said.

Hague also said Hamas “bears principal responsibility” for initiating the violence, but made clear the diplomatic risks of an Israeli escalation. “A ground invasion is much more difficult for the international community to sympathize with or support,” he said.

A ground operation would carry grave risks, given the likelihood of heavy casualties on both sides. The Israeli offensive into Gaza four years ago left hundreds of civilians dead, drawing fierce international condemnation and war crimes accusations.

Israel says its intelligence and technology have been perfected since then to minimize civilian casualties. But Gaza’s crowded urban landscape makes it all but impossible to avoid them altogether, as Sunday’s attack in Gaza City illustrated.

“In this case, you can’t avoid collateral damage if they position the rockets in densely populated areas, in mosques, school yards,” said Israeli Vice Premier Moshe Yaalon. “We shouldn’t be blamed for the outcome.”

Avihai Mandelblit, a recently retired chief advocate general in the Israeli military, said that from a legal perspective, “there’s no immunity to anyone if you put weapons inside of civilian infrastructure.”

But he acknowledged the sight of dead civilians could create a public relations debacle for Israel. “As more civilians will get hurt, the legitimacy clock is going to click faster to end this operation,” he said.

Obama said he had been in touch with Netanyahu as well as the leaders of Egypt and Turkey as international attempts to broker a cease-fire continued. Egypt, which often serves as a mediator between Israel and Hamas, has taken a leading role in the efforts.

Israeli TV stations said an Israeli envoy traveled to Cairo on Sunday, and was returning to Israel with details of cease-fire proposals. Channel 2 TV, citing American diplomats, said Netanyahu’s personal envoy, Yitzhak Molcho, would be headed to Washington in the coming days.

Hamas officials said their supreme leader, Khaled Meshaal, also held talks with Egyptian President Mohammed Morsi, and that Turkish Foreign Minister Ahmet Davutoglu was to visit Gaza on Tuesday.

Israel and Gaza’s militant Hamas rulers remain far apart on any terms for a halt to the bloodshed.

Hamas is linking a truce deal to a complete lifting of the border blockade on Gaza imposed since Islamists seized the territory by force. Hamas also seeks Israeli guarantees to halt targeted killings of its leaders and military commanders. Israeli officials reject such demands.

They say they are not interested in a “time-out,” and want firm guarantees that militant rocket fire into Israel will end. Past cease-fires have been short lived.