JERUSALEM — Israeli troops pushed into a heavily populated area of Gaza City from the south on Sunday in fierce fighting as senior Israeli officials said that they believed the Hamas military wing was beginning to crack and that Hamas leaders inside Gaza were looking for a cease-fire.

Prime Minister Ehud Olmert of Israel told the nation Sunday that it was "getting close to achieving the goals it had set for itself" but that "more patience, determination and effort is still demanded."

The comments by Olmert, made during the public part of the regular Sunday cabinet meeting in Jerusalem, were broadcast to an Israeli public that generally supports the war against Hamas in Gaza but is unquiet about how and when it will end.

In his remarks, Olmert gave no time frame for the conflict but said that Israel "must not miss out, at the last moment, on what has been achieved through an unprecedented national effort."

General Amos Yadlin, the head of army intelligence, and Yuval Diskin, the head of Israel's Shin Bet internal security agency, said that "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," Oved Yehezkel, the cabinet secretary, told reporters. The Israelis said this view inside Gaza was a contrast to the "unyielding stands" of the exiled Hamas leadership in Damascus, in particular of Khaled Meshal, the group's 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 were quoted as saying.

Another senior Israeli security official, speaking on the condition of anonymity, said in an interview 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.

Israel and Washington are trying to secure a deal that would mean a Hamas commitment to stop firing rockets into Israel and an Egyptian commitment to block the tunnels into Gaza that are used to resupply Hamas. In return, Israel would agree to a cease-fire and to the opening up of its crossings into Gaza for goods and fuel, as well as the opening of the Rafah crossing into Egypt under European Union supervision.

If the Egyptian angle fails, Israeli officials said, the military is likely to go to a "third stage" of the war against Hamas in Gaza, with reserve troops thrown into the battle. An expansion of the war would likely mean Israeli troops moving into southern Gaza to take a strip of land at least 450 meters, or 1,500 feet, wide inside the Egyptian border. Israel has been bombing the area to try to destroy smuggling tunnels between Gaza and Egypt, the destruction of which is one of the main aims of the war.

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

The fighting Sunday started before dawn in the Sheikh Ajleen neighborhood, in the southwestern edge of Gaza City, as Israeli troops moved north from Netzarim, where they had earlier cut the Gaza Strip in two. In a sense, the Israelis opened a second front from the south, because they have also been fighting on the northern outskirts of Gaza City.

At least 20 Palestinians had been killed by midday, when fighting ebbed somewhat, Palestinian hospital officials said. The fighting was described as fierce and close to Al Quds Hospital in Sheikh Ajleen. Meanwhile, Palestinian medical workers said that three militants were killed by airstrikes.

Hamas and Islamic Jihad said they had ambushed the Israelis, but there was no immediate indication of whether any Israeli soldiers had been hurt.

Israeli warplanes also bombed a region along the Egypt-Gaza border, near the town of Rafah, where smuggling tunnels are located.