JERUSALEM — Israel kept up a steady bombardment of the Gaza Strip from the air and sea on Friday as the United States offered to broker a cease-fire that would end an offensive launched to quell rocket fire from the territory.

The death toll from the Israeli attacks continued to climb, with the United Nations reporting 114 people killed, of whom 88 were civilians, including 30 children and 17 women. Another 850 people were injured, more than two-thirds of them women and children, according to the Gaza Health Ministry.

In Israel, a man was seriously wounded when a rocket fired from Gaza hit a gas station in the southern port city of Ashdod. A woman was injured in a rocket strike that hit homes in the southern city of Beersheba.

A rocket fired by militants in Lebanon landed in northern Israel in an open area, causing no casualties or damage, the army said. Israeli troops responded with artillery fire.

The Israeli army said that more than 600 rockets and mortar rounds had been launched from Gaza since the start of the offensive against the militant Islamist group Hamas and allied factions. They have caused only limited damage and no deaths.

In a phone call Thursday with Prime Minister Benjamin Netanyahu, President Barack Obama “reaffirmed Israel’s right to defend itself” against rocket attacks but “expressed concern about the risk of further escalation and emphasized the need for all sides to do everything they can to protect the lives of civilians and restore calm,” the White House said in a statement.

“The United States remains prepared to facilitate a cessation of hostilities, including a return to the November 2012 cease-fire agreement,” the statement added, referring to a truce that ended a previous Israeli offensive against Hamas.

Dan Shapiro, the American ambassador to Israel, told Israel Radio that Secretary of State John Kerry had held contacts with Egypt, Qatar and other governments in the region to help mediate an end to the fighting.

“We are prepared to work with any country that can influence Hamas to stop the launching of missiles and rockets and restore calm,” he said.

Shapiro said Egypt could play a central role in cease-fire contacts “but the situation is more complicated” because of the change in leadership in Cairo, where President Abdel-Fattah el-Sissi has taken a hostile stance toward Hamas.

Netanyahu appeared to rule out any immediate movement toward a cease-fire on Friday.

“No international pressure will prevent us from operating with full force against a terrorist organization that calls for our destruction,” he said. “We will strike hard at all those who try to attack us.”

In fresh Israeli airstrikes early Friday targeting homes of suspected militants, five members of a single family were reported killed, including a 7-year-old girl, when a multistory building was bombed in Rafah in the southern Gaza Strip. More than a dozen people were reported injured when the building collapsed.

The army said it had targeted the home of a senior Hamas operative in the Rafah area who had participated in rocket attacks on Israel and Israeli forces, though it was unclear whether it was referring to the same building.

A 50-year-old man and a 10-year-old girl were reported killed in other attacks, which the army said also targeted tunnels, military compounds, training sites and long-range rocket launchers.

At the al Bureij refugee camp, a vehicle belonging to the local municipality was struck, killing two employees and wounding three bystanders, one of them a 9-year-old boy who died later of his wounds, the U.N. said.

Since the start of the Gaza offensive, more than 1,000 targets in the densely populated coastal enclave have been hit with more than 2,000 tons of explosives, the army said.

The United Nations office for the coordination of humanitarian affairs said that at least 75 homes of suspected militants in the Gaza Strip had been specifically targeted and totally destroyed. Overall, 542 homes had been destroyed or severely damaged, displacing 3,250 people.

Five health facilities had suffered damage from airstrikes nearby, and hospitals treating the injured were suffering from severe shortages of medical supplies and fuel needed to operate backup generators, the U.N. agency said.

Hamas appeared unbowed by the Israeli onslaught. In an interview with Hamas television, Mahmoud Zahar, a top leader of the group in Gaza, said that “we are ready to continue with this battle, on an even bigger scale then it is now, for months.”

He said that Hamas would agree to a cease-fire only if its conditions were met, including lifting an Israeli and Egyptian “siege” on Gaza and freeing former prisoners rearrested by Israel in recent weeks in the West Bank. They were detained during a crackdown on Hamas after the kidnapping and killing last month of three Israeli teenagers in the West Bank, which Israel blamed on the militant group.

In nationally televised remarks Thursday, Netanyahu said that “as the campaign continues, we will deepen the damage” to Hamas, and he urged Israelis to show “fortitude” in what he called a “difficult battle” ahead.

Israel’s Iron Dome missile defense system has intercepted more than 120 rockets fired from Gaza at Israel’s population centers, according to an army count, and air-raid sirens have warned people to take shelter, preventing casualties. Many rockets have landed in unpopulated areas.

Greenberg is a McClatchy special correspondent. Email: jgreenberg@mcclatchydc.com