JERUSALEM — Diplomatic efforts accelerated Tuesday to end the deadly confrontation between Israel and Palestinian militants in Gaza, as the United States sent Secretary of State Hillary Rodham Clinton to the Middle East and Egypt’s president and his senior aides expressed confidence that a cease-fire was close.

The diplomatic moves to end the nearly week-old crisis came on a day of some of the most intense violence yet. Militants in Gaza fired a barrage of rockets toward the southern Israeli city of Beersheba and longer-range rockets toward Tel Aviv and Jerusalem, but neither main city was struck. Israeli forces responded with an aerial assault on several targets including a suspected launching site near Gaza’s Al Shifa hospital, which killed at least nine people. A delegation visiting from the Arab League postponed a news conference at the hospital because of the Israeli assault, as wailing ambulances brought victims in, some of them decapitated.

Senior Egyptian officials in Cairo said Israel and Hamas were “very close” to a cease-fire agreement that could be announced within hours. “We have not received final approval but I hope to receive it any moment,” said Essam el Haddad, President Mohamed Morsi’s top national security adviser.

The announcement of Mrs. Clinton’s active role in efforts to defuse the crisis added a strong new dimension to the multinational push to avert a new Middle East war. Israel has amassed thousands of soldiers on the border with Gaza and has threatened to invade the crowded Palestinian enclave for the second time in four years to stop the persistent rockets that have been lobbed at Israel.

Mrs. Clinton, who accompanied President Obama on his three-country Asia trip, left Cambodia on her own plane immediately for the Middle East. She was en route to Jerusalem to meet with Prime Minister Benjamin Netanyahu of Israel, then head to the West Bank to meet with Palestinian leaders and finally to Cairo to consult with Egyptian officials.

In Cairo, President Morsi added to the atmosphere of guarded optimism. The official Middle East News Agency quoted him as saying Israel’s “aggression” against Gaza would end, and Egyptian-mediated efforts would produce “positive results” in several hours.

The decision to dispatch Mrs. Clinton dramatically deepens the American involvement in the crisis. Mr. Obama made a number of late-night phone calls from his Asian tour to the Middle East on Monday night that contributed to his conclusion that he had to become more engaged and that Mrs. Clinton might be able to accomplish something.

With Ban Ki-moon, the United Nations secretary general, also in Israel on Tuesday, a senior official in the prime minister’s office said Israel had decided to give more time to diplomacy before launching a ground invasion into Gaza. But Israel has not withdrawn other options.

“I prefer a diplomatic solution,” Mr. Netanyahu said in a statement at the start of a meeting in Jerusalem with the German foreign minister, Guido Westerwelle. “I hope that we can get one, but if not, we have every right to defend ourselves with other means and we shall use them.

“As you know, we seek a diplomatic unwinding to this, through the discussions of cease-fire,” Mr. Netanyahu added. “But if the firing continues, we will have to take broader action and we won’t hesitate to do so.”

Intensifying the pressure on Hamas after a day of heavy rocket fire out of Gaza against southern Israel, the Israeli military said on Tuesday afternoon that it had distributed leaflets over Gaza instructing the Palestinian residents in several areas to evacuate their homes immediately, “for your safety,” and to move toward defined zones in central Gaza City. That seemed intended to signal that plans for a ground operation were imminent should the cease-fire talks fail

About three hours before Mr. Ban was scheduled to meet Mr. Netanyahu in Jerusalem, sirens sounded across the city in the early afternoon announcing an incoming rocket from Gaza. The military wing of Hamas said it had fired at the city. The rocket fell short, landing harmlessly in the West Bank just south of Jerusalem, and the military said it landed on open ground near a Palestinian village.

The rocket attack on the city, which is holy to Jews, Muslims and Christians, was the second in less than a week. On Friday, a rocket landed in a similar location, the police said.

The Israeli military said its air force had struck Tuesday morning at 11 Palestinian squads involved in planting explosives and firing rockets, as well as underground rocket launchers and a store of weapons and ammunition. The military said it had also used tank shells and artillery fire against unspecified targets in Gaza.

The Health Ministry in Gaza said the Palestinian death toll had climbed by late Tuesday morning to 112, roughly half of the dead civilians, including children. Three Israelis died in a rocket attack last week.

After an Asian summit dinner in Phnom Penh on Monday night, Mr. Obama called President Morsi to discuss the situation, then spoke with Mr. Netanyahu and called Mr. Morsi back. He was up until 2:30 a.m. on the phone, the White House said. He consulted with Mrs. Clinton repeatedly on the sidelines of the Asian summit meetings on Tuesday.

“This morning, Secretary Clinton and the president spoke again about the situation in Gaza, and they agreed that it makes sense for the secretary to travel to the region, so Secretary Clinton will depart today,” said Benjamin Rhodes, a deputy national security adviser to Mr. Obama. “Her visits will build on the engagement that we’ve undertaken in the last several days.”

Mr. Rhodes said, “Any resolution to this has to include an end to that rocket fire” by Hamas militants on Israeli communities, but “the best way to solve this is through diplomacy.”

He added, “It’s in nobody’s interest to see an escalation of the military conflict.”

Mrs. Clinton will not meet with Hamas representatives on her trip, but with the Palestinian leadership in the West Bank, which is at odds with the Hamas rulers of the Gaza Strip. “We do not engage directly with Hamas,” Mr. Rhodes said.

Instead, Mr. Obama is focused on leveraging Egypt’s influence with Hamas to press for a halt to the rocket attacks. “We believe Egypt can and should be a partner in achieving that outcome,” Mr. Rhodes said.

Mr. Rhodes reaffirmed that the United States supports Israel’s right to defend itself and said Mr. Obama did not ask Mr. Netanyahu to hold off a ground incursion into Gaza.

In Jerusalem, an official in the prime minister’s office said the country’s top nine ministers, who make up the inner security cabinet, held discussions late into the night on the state of the diplomatic efforts and Israel’s military operation in Gaza. The goal of the operation, Israel says, is to end years of rocket fire by Gaza militants against southern Israel.

Egypt has been brokering efforts, with American involvement, for a cease-fire. “What is on the table is not there yet,” the official said, referring to Israel’s demands for an end to the threat of rocket fire. “It does not bring about what we need.”

Gaza militants fired more than 70 rockets in heavy barrages at southern Israel on Tuesday morning, and 8 of them struck in densely populated areas. A soldier was wounded in one rocket attack, according to the military. Houses were hit in Beersheba and Netivot, causing damage but no injuries. Other rockets hit a car and damaged a bus in Beersheba. The passengers had disembarked on hearing a warning siren and escaped harm. About 27 rockets aimed at populated areas were intercepted by Israel’s Iron Dome anti-rocket missile system and others landed in open ground.

Separately, an Israeli man attacked an Israeli security guard outside the American Embassy in Tel Aviv with an ax and a knife on Tuesday, injuring the guard. The guard fired warning shots, apprehended the assailant and handed him over to the Israeli police, according to a police spokesman who described the episode as a “criminal incident” unrelated to the current security situation.

Tens of thousands of Israeli reserve soldiers have been mobilized and troops and tanks have massed along the border with Gaza, ready to go in case the order is given.

So far Israel has carried out its campaign from the air, pounding more than 1,000 targets in Gaza, including long-range rocket launchers and weapons stores. Gaza militants have fired more than 800 rockets at Israel and several have reached as far north as Tel Aviv.

There has been no apparent letup in Israeli airstrikes.

The Israeli military said on Tuesday that its warplanes had hit 100 targets overnight. One was the main branch of the National Islamic Bank, which Hamas opened in 2010. Witnesses said two rockets fired from an Apache helicopter hit the bank, on the ground floor of an apartment building, igniting a large fire that injured seven people who lived nearby.

Late Monday night, a bomb dropped from an Israeli F-16 warplane crashed into a house in the northern town of Beit Lahiya, killing Fowad Khalil Hijazi, 46, and two of his children, Mohamed, age 3, and Suhaib, 2, according to officials at Kamal Adwan Hospital there.

Their mother was in critical condition. At the Hijazi home, neighbors were combing through the crater left by the bomb, pulling out remnants of the explosives and laying them on a mat, and packing jars of olives and other food into a storage container.

Among the latest casualties were Akram Marouf, who died in a drone attack in Beit Lahiya; a 15-year-old boy who was hunting birds in an open space in Beit Lahiya when he was struck; and Mahmoud Al Zahar, felled by an airstrike in central Gaza City.

Mrs. Clinton’s trip comes as she is preparing to step down as secretary of state, presenting her with a delicate late test after four years in which Mr. Obama’s administration has failed to achieve the broader peace it once sought in the region.

With the president’s re-election behind him, Mrs. Clinton plans to resign around the time of the second inauguration on Jan. 20. Aides said she would stay until a successor could be confirmed as long as the confirmation process does not drag too long into the new year.

The abrupt change in plans here underscored the challenges for Mr. Obama as he tries to reorient American foreign policy away from its dominant focus on the Middle East and more toward the Pacific-Asia region Even as he chose Southeast Asia as the destination for the first overseas trip after winning a second term, Mr. Obama has found himself drawn every day into the deadly dispute consuming the Middle East.