Israeli troops pushed into a heavily populated area of Gaza City from the south early on Sunday in what the army and locals described as fierce fighting. In Jerusalem, Israeli Prime Minister Ehud Olmert told the nation that Israel is “getting close to achieving the goals it set for itself,” but that “more patience, determination and effort is still demanded.”

Speaking during the regular Sunday cabinet meeting, Mr. 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.”

Mr. Olmert and his two top cabinet ministers, Defense Minister Ehud Barak and Foreign Minister Tzipi Livni, are 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 on the ground unless they are constantly moving.

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

The fighting started before dawn on Sunday in the Sheikh Ajleen neighborhood, in the southwestern edge of Gaza City, with Israeli troops moving from Netzarim, where they had earlier cut the Gaza Strip in two.

At least eight people died — four fighters, two women and two children, according to Palestinian hospital officials. The fighting was described as fierce and close to Al Quds Hospital in Sheikh Ajleen. Hamas and Islamic Jihad said they had ambushed the Israelis, but there was no immediate indication that any Israeli soldiers had been hurt.

Palestinian medical workers said that another three militants were killed by air strikes.

Three rockets were fired from Gaza at Israel on Sunday morning, Israeli Army radio said. Two exploded near Beersheva, injuring several people. The third hit empty land.

On Saturday, the Hamas political director in exile, Khaled 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.”

Still, a Hamas delegation was holding talks in Egypt on an Egyptian plan for a truce, and a senior Israeli defense official, Amos Gilad, is expected to make a second trip to Cairo in the next day or so, Israeli officials said.

Israel and Washington are trying to secure agreement on a deal 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 a resupply of Hamas. In return, Israel would agree to a cease-fire and the opening up of its crossings for goods and fuel into Gaza and the opening of the Rafah crossing into Egypt, with European Union supervision.

Also on Sunday, Israeli officials told the Haaretz newspaper that a military investigation concluded that an Israeli mortar shell, despite having a guidance system, was 30 yards off its target when it hit near a United Nations school, killing as many as 43 Palestinians.

Two other mortars hit their target, a Hamas cell that had fired mortars at Israeli troops, killing at least two of the fighters, the army said.

An initial effort to use an even more precise weapon failed because of a malfunction, the army told Haaretz. The army appeared to drop its contention that the Hamas mortars came from within the school compound, but said instead that the mortars were from an adjacent area.

On Saturday, United Nations relief operations resumed after a daylong suspension prompted by fears for the safety of the drivers. On Thursday, a United Nations driver was killed and two others were wounded from what the agency said was Israeli fire. Israel issued a statement on Saturday saying it was certain that the shooting had not come from its forces, adding that the drivers were treated in an Israeli hospital. It also redoubled its assurances to the United Nations on holding its fire around aid convoys.

Tank and artillery fire pounded Gaza all night and day, with plumes of black smoke visible especially in the eastern part of Gaza City. A tank shell landed outside the home of a family in Jabaliya, northeast of the city, killing eight members of the same family who were sitting outside, hospital officials said.

The Palestinian president, Mahmoud Abbas, whose Fatah faction opposes Hamas, was in Cairo pressing a call for a cease-fire, and he discussed with President Hosni Mubarak of Egypt the idea of international troops along the Gaza-Egypt border. Hamas representatives were also there, but the plan, also urged by the French, seemed to be losing steam. Egypt is unenthusiastic about the presence of foreign troops on its soil, while Hamas is unwilling to have the troops inside Gaza.

More focus was being placed on technical assistance to the Egyptians to help them block and destroy the smuggler tunnels that help Hamas stay lethal.

Both Israel and Hamas rejected a United Nations Security Council resolution on Friday calling for a cease-fire. And the actions of both on Saturday made their resolve to keep fighting manifest.

More long-range rockets hit Israel on Saturday, including two in open areas in Ashdod, a city of 200,000 on the way to Tel Aviv. A tank shell landed outside the home of a family in Jabaliya, northeast of Gaza City, killing eight members of the same family who were sitting outside, hospital officials said

On Saturday, Israel warned Gaza residents that it was preparing the next phase of its war against Hamas — a deeper ground force operation — as diplomatic efforts to end the 16-day assault and Hamas rocket fire into Israel faltered.

In Gaza City as well, residents reported getting phone calls that said: “We are going to intensify the military strike against Hamas. Our intention is not to harm civilians. If you live near Hamas, evacuate.”

Leaflets were dropped addressed to “the residents of Gaza,” saying that the Israeli military had in recent days warned residents of the southern city of Rafah of “an imminent operation,” and asking them to evacuate their homes for their safety.

“The fact that the residents of Rafah abided by the orders,” the leaflets continued, “has protected those who had nothing to do with the fight. The Israel Defense Forces will intensify shortly its directed operation against tunnels, weapon storehouses and members of terrorist groups all over Gaza. For your safety and that of your family you are asked to stay away from terrorist elements and from places where terrorist operations occur. Please continue abiding by our orders.”

Red Cross workers said their telephones were flooded with calls from residents of the Beach refugee camp who had received large numbers of the calls and leaflets. The callers wanted to know if they should evacuate their homes, and if so, to where.

A Beach camp car mechanic named Hamdi Eki, 47, was asked why he did not leave after receiving such a call. “I have nine children,” he said. “Where can I go? I prefer to die at my own house.”

Some Beach camp residents did leave but ended up in other neighborhoods or camps that had received similar warnings.

Israel has come under increasing international criticism for the growing number of civilian casualties of this war and for complicating efforts by aid and rights groups to help those caught in the cross-fire. Israel says Hamas fighters hide consciously among civilians, in mosques and schools and under clinics.

Fred Abrahams, a senior researcher at Human Rights Watch, who has studied both the Kosovo and Lebanon conflicts, said he was concerned that Israel was not paying enough attention to international legal requirements for “distinction and proportionality — first, to distinguish between combatants and civilians, and second, whether an attack will have a disproportionate effect on the civilians in the area.”

Even if a target is legitimate, he said, “you can’t drop a 500-pound bomb in an area crowded with civilians.”

This was also the first conflict he could remember when civilians could not flee the war zone. Gaza’s borders are shut both to Israel and to Egypt, and civilians, he said, “are fish in a barrel.”

“Our conclusions are preliminary, but evidence is suggesting serious violations of the laws of war, which require investigation,” Mr. Abrahams said.

That is also true of Hamas, he said. “We need to know more about what Hamas is doing on the ground,” he said. “For example, we know Hamas has stored weapons in mosques, so when Israel targets a mosque, we don’t scream war crime.”

Regarding force protection, he said it “must be balanced by distinction and proportion.”

“A violation by Hamas shooting from a mosque or school doesn’t give the Israeli Army carte blanche to return fire in the name of force protection with everything and anything it has,” he added.

Human rights groups are also concerned about the Israeli use of white phosphorous, which creates smoke on a battlefield, at low altitudes or crowded areas, because it can burn like a kind of napalm.