On the 17th day of the war against Hamas, Israel said its ground forces called in a series of air strikes after troops pushed into a heavily populated area of Gaza City from the south on Sunday in fierce fighting that continued on Monday. Senior Israeli officials said Sunday for the first time in the war that they believed that the Hamas military wing was beginning to crack and that Hamas leaders inside Gaza were looking for a cease-fire.

But news reports on Monday said Hamas militants fired as many as 10 missiles out of Gaza into southern Israel without causing casualties.

Overnight, the Israeli military said, its warplanes carried out fewer strikes than on some previous nights.

By midday Monday, the Israeli military said its warplanes had struck 25 targets including a mosque said to be used to store Hamas rockets and mortars. Facing accusations that its offensive has created a humanitarian crisis in Gaza, the military said 165 truckloads of aid were being allowed into Gaza.In Jerusalem on Sunday, Israel’s prime minister, Ehud Olmert, told the nation that Israel was “getting close to achieving the goals it set for itself,” but that “more patience, determination and effort are still demanded.”

Mr. Olmert was speaking in the public part of the regular Sunday cabinet meeting, and his words were broadcast to an Israeli populace that supports the war against Hamas in Gaza but is nervous about how and when it will end.

Mr. Olmert gave no time frame, but said Israel “must not miss out, at the last moment, on what has been achieved through an unprecedented national effort.”

The Israeli military said on Monday that warplanes attacked five Hamas operatives along with weapons caches, tunnels and other targets, while Israeli gunboats fired from the sea. Israeli officials also said Sunday that the military had been sending reserve units into Gaza since Thursday. They did not specify the number of reservists. The announcement appeared aimed at adding pressure on Hamas but it also raised the possibility of an expansion in the conflict, which began Dec. 27. It was not clear Monday whether the so-called “third phase” of the war had been approved by Israeli leaders.

On Monday Egypt planned to convene negotiations aimed at a cease-fire in Gaza, where the Israeli military assault to silence rocket fire and tunneling by Hamas and other militants opposed to Israel’s existence has wrought extensive death and destruction.

Nearly 900 people have been killed, according to Palestinian Health Ministry officials. Thirteen Israelis have been killed, Israel has said.

European diplomats involved in the Egypt negotiations said Sunday that the next 48 hours would be crucial for Israel to decide if a durable cease-fire can be achieved.

The Israeli cabinet secretary, Oved Yehezkel, told reporters that in the cabinet meeting the heads of army intelligence, Maj. Gen. Amos Yadlin, and of the Shin Bet security service, Yuval Diskin, said, “It is the inclination within Hamas to agree to a cease-fire, given the harsh blow it received and given the absence of accomplishment on the ground.”

The Israelis said this view inside Gaza was a contrast to the “unyielding stands” of the exiled Hamas leadership in Damascus, Syria, in particular Khaled Meshal, the political director. But Hamas “is not expected to wave a white flag” and is reserving rockets and weaponry to fire at the end of the conflict, the intelligence chiefs said.

Another senior Israeli security official said that Israeli soldiers had “confirmed through their sights” the killing of 300 Hamas and Islamic Jihad fighters on the ground in Gaza, and that Hamas units were making mistakes and fighting without clear direction.

“I can say with a high level of confidence that for two days, what we have been hearing repeatedly is that Hamas inside Gaza is eager — eager — to achieve a cease-fire,” said the senior official, who spoke on condition of anonymity because of the issue’s delicate nature. “This is as opposed to the leadership in Damascus that is willing to fight to the last Palestinian.”

The Israelis were clearly all pushing a concerted message, but no official provided details on how Israel supported its assertion. It was impossible to get a response from Hamas leaders in Gaza, because they were in hiding from Israeli military strikes.

On Saturday, the Hamas political director in exile, Mr. Meshal, said in Damascus that Hamas would not consider a cease-fire until Israel ended the assault and opened all crossings into Gaza. He said that the ferocity of the Israeli campaign had crossed the line and called it a “holocaust,” adding, “You have destroyed the last chance for negotiations.”

Israel and the United States are trying to secure agreement on a deal brokered by Egypt that would mean a Hamas commitment to stop all rocket firing into Israel and an Egyptian commitment to block smuggling tunnels into Gaza, to stop the resupplying of Hamas with weaponry and cash. In return, Israel would agree to a cease-fire and the opening of its crossings into Gaza for goods and fuel and the opening of the Rafah crossing into Egypt, with European Union supervision.

Tony Blair, the former British prime minister and now an international envoy to the Palestinians, said in an interview that “the only way this is going to stop is if there is a genuine plan to end the smuggling into Gaza and a genuine plan to open the crossings.”

Mr. Blair will be in Cairo on Monday, as will a senior Israeli Defense Ministry official, Amos Gilad. A Hamas delegation is already in Cairo talking to the Egyptians through the intelligence chief, Omar Suleiman. If the Egyptian effort fails, Israeli officials said, the military is likely to go to a “third stage” of the war against Hamas in Gaza, with the reserve troops thrown into the battle.

An expansion of the war would most likely mean Israeli troops moving into southern Gaza, to take a strip of land at least 500 yards wide inside Gaza at the Egyptian border. Israel has been bombing the area to try to destroy smuggling tunnels between Gaza and Egypt.

Mr. Olmert and his two top cabinet ministers, Defense Minister Ehud Barak and Foreign Minister Tzipi Livni, were reported to disagree about the best way to win the war and consolidate Israeli gains. But they are under pressure from the army to decide on whether to expand the war or end it, in part because the soldiers become easier targets unless they are constantly moving.

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There was a new development on Sunday in the investigation into one of the deadliest attacks so far — an Israeli mortar strike near a United Nations school on Tuesday that killed up to 43 Palestinians. The newspaper Haaretz reported that a military investigation had concluded that two Israeli shells hit a Hamas mortar unit that had fired first, but that an errant Israeli shell hit near the school.

The army later rebutted the article, saying its initial inquiry showed “mortars were fired from within the school” at Israeli forces nearby, “and those forces returned fire.”

United Nations officials have denied that any Palestinian fighters were in the school grounds and called for an independent international investigation, and the army had earlier gone back and forth about whether the Hamas mortars were fired within the school or near it.