JERUSALEM — Efforts to agree on a cease-fire between Israel and Hamas intensified Tuesday, but the struggle to achieve even a brief pause in the fighting emphasized the obstacles to finding any lasting solution.
On the deadliest day of fighting in the week-old conflict, Secretary of State Hillary Rodham Clinton arrived hurriedly in Jerusalem and met with Prime Minister Benjamin Netanyahu of Israel to push for a truce. She was due in Cairo on Wednesday to consult with Egyptian officials in contact with Hamas, placing her and the Obama administration at the center of a fraught process with multiple parties, interests, and demands.
Officials on all sides had raised expectations that a cease-fire would begin around midnight, followed by negotiations for a longer-term agreement. But by the end of Tuesday, officials with Hamas, the militant Islamist group that governs Gaza, said any announcement would not come at least until Wednesday.
The Israelis, who have massed tens of thousands of troops on the Gaza border and have threatened to invade for a second time in four years to end the rocket fire from Gaza, never publicly backed the idea of a short break in fighting. They said they were open to a diplomatic accord but were looking for something more enduring.
‘‘If there is a possibility of achieving a long-term solution to this problem through diplomatic means, we prefer that,’’ Netanyahu said before meeting with Clinton at his office. ‘‘But if not, I’m sure you understand that Israel will have to take whatever actions necessary to defend its people.’’
Clinton spoke of the need for ‘‘a durable outcome that promotes regional stability and advances the security and legitimate aspirations of Israelis and Palestinians alike.’’ It was unclear whether she was starting a complex task of shuttle diplomacy or whether she expected to achieve a pause in the hostilities and then head home.
The diplomatic moves came as the antagonists on both sides stepped up their attacks. Israeli aerial and naval forces assaulted several Gaza targets in multiple strikes, including a suspected rocket-launching site near Al Shifa Hospital. That attack killed more than a dozen people, bringing the total fatalities in Gaza to more than 130 — roughly half of them civilians, the Gaza Health Ministry said.
A delegation visiting from the Arab League canceled a news conference at the hospital because of the Israeli aerial assaults as wailing ambulances brought victims in, some of them decapitated.
The Israeli assaults carried into early Wednesday, with multiple blasts punctuating the otherwise darkened Gaza skies.
Militants in Gaza fired a barrage of at least 200 rockets into Israel, killing an Israeli soldier — the first military casualty on the Israeli side since the hostilities broke out. The Israeli military said the soldier, identified as Yosef Fartuk, 18, had died from a rocket strike that hit an area near Gaza. Israeli officials said a civilian military contractor working near the Gaza border was also killed, bringing the number of fatalities in Israel from the week of rocket mayhem to five.
Other Palestinian rockets hit the southern Israeli cities of Beersheba and Ashdod, and longer-range rockets were fired at Tel Aviv and Jerusalem. Neither main city was struck. One Gaza rocket hit a building in Rishon LeZion, just south of Tel Aviv, injuring one person and wrecking the top three floors.
Senior Egyptian officials in Cairo said Israel and Hamas were ‘‘very close’’ to a cease-fire agreement. ‘‘We have not received final approval, but I hope to receive it any moment,’’ said Essam el-Haddad, President Mohammed Morsi’s top foreign affairs adviser.
Foreign diplomats who were briefed on the outlines of a tentative agreement said it had been structured in stages — first, an announcement of a cease-fire, followed by its implementation for 48 hours. That would allow time for Clinton to involve herself in the process on the ground here and create a window for negotiators to agree on conditions for a longer-term cessation of hostilities.
US officials said Washington was betting on the pragmatism of Egypt’s new president, a former leader of the Muslim Brotherhood, the movement with which Hamas is affiliated.
While President Obama publicly stressed Israel’s right to self-defense because of domestic political concerns, officials said the administration also had decided to take an understanding approach to Morsi’s need to denounce Israel in order to appeal to his domestic audience.
‘‘We know that the Egyptians have their domestic politics as well,’’ one US official said, and each president understood the other’s political context. ‘‘But they both agree that this nonsense can’t go on.’’
Officials of Morsi’s government acknowledged that the Gaza battle had put them in a bind. As Egypt’s first democratically elected president, Morsi must respond to a public deeply angry at Israel and eager to rally behind the Palestinians. ‘‘But if he responds fully to public opinion, he risks what we have been trying to do for peace and stability in the region,’’ a senior official said, speaking on the condition of anonymity.
Indeed, despite the Egyptian government’s caustic statements about Israel and noisy solidarity with Hamas, several US officials said Morsi and the new Islamist government needed no encouragement in its efforts to push for an end to not only the Israeli bombing but also Hamas’s missile fire.
But Israel wants guarantees that Egypt will stop the flow of arms into Gaza from Sinai, and that seems a tall order. Egypt has been unable to control Sinai and would not want to be seen in the role of Israeli enforcer. Egypt is hoping Hamas will restrain itself on missile imports, but it is far from clear that Hamas wants to or can, given the range of forces in Gaza vying for power, including the Iranian-backed Islamic Jihad.
Within Hamas itself, there are divisions and fractured views on the truce negotiations. In Gaza on Tuesday, Fawzi Barhoum, the Hamas spokesman, said that ‘‘we hold absolutely no hope of Hillary Clinton’’ helping to resolve the conflict.
Obama, who was in Asia, had found himself repeatedly on the phone with Middle Eastern leaders in recent days and decided that Clinton, who also spoke to a dozen of her counterparts here, could make the difference in setting a cease-fire.
Netanyahu’s calculations are numerous. He has an election looming in January, and agreeing to stop his operation in Gaza could be risky if rocket fire resumed. But sending troops into Gaza poses perhaps even more risks.
‘‘The Israeli government will face its voters without any tangible achievement in hand to show,’’ Nahum Barnea, a columnist for the Yediot Aharonot newspaper, wrote Tuesday. He said that he did not believe Netanyahu had begun this operation with electoral considerations in mind, but that ‘‘the deliberations about ending it are deeply affected by political calculations.’’
Netanyahu is also contending with a radically altered Middle East, and while he says that protecting his people is not dependent on who is in power in Egypt or Turkey, a reduced military operation and fewer civilian casualties in Gaza would make relations with both countries less difficult.