Despite mounting diplomatic pressure to end its offensive in Gaza, Israel’s military onslaught unfolded for an 11th day on Tuesday amid reports that it had struck near a United Nations school, killing at least 30 people among hundreds who had sought refuge from the fighting.

The United Nations said 30 people had been killed in the strike near the school, which is in Jabaliya in Gaza. Another 55 were injured, five of them critically, it said. Three hospital officials at Shifa hospital, where some of the wounded were taken, said at least 42 people were killed and that the number of likely was likely to rise. Those killed included women and children.

It was unclear whether the strike came from Israeli tank or rocket or other artillery, and the numbers could not be independently confirmed. In the past, there have been concerns that death tolls have been exaggerated for political purposes.

Reuters reported that two shells exploded outside the school, spraying shrapnel inside and outside the building where Palestinians had sought refuge. It cited medical sources at two hospitals, who said the dead were either people taking shelter in the school or local residents.

Israel has been criticized in the past for the inaccuracy of its shelling. The Israeli Army has repeatedly emphasized that its operation is not aimed at Gaza’s residents, amid sensitivity to deep opposition worldwide to the toll on civilians in Gaza.

But parts of Gaza, a narrow coastal strip with a population of 1.5 million, are among the most densely crowded areas in the world, and artillery and tank fire can easily cause collateral damage. In November 2006, Israel all but stopped firing tank and artillery shells into Gaza after 18 Palestinian civilians, most from one family, were killed by Israeli shells that missed their target and hit a row of houses in Beit Hanoun.

In another strike, during its conflict in Lebanon in July 2006, Israel suspended air attacks in southern Lebanon for 48 hours after one of its raids on the southern town of Qana left dozens of civilians, many of them children, dead.

The United Nations said that the Israeli army been given all the coordinates of United Nations facilities in Gaza, and that the schools and other facilities were all clearly marked.

Separately, the Israeli Army would not confirm reports that its ground troops were pushing further south through Gaza toward Khan Yunis, the beleaguered territory’s second city.

A rocket fired by Hamas from Gaza fell in the Israeli town of Gadera, less than 20 miles south of Tel Aviv and the furthest north that any of the hundreds of missiles fired from Gaza has yet struck since the Israeli offensive began, the Israeli Army said. Amid the fighting within Gaza, four Israeli soldiers were killed by shells from their own tanks, the first Israeli deaths from so-called “friendly fire” in the conflict.

As the diplomatic pressure on Israel intensified, French President Nicolas Sarkozy, who is touring the region in quest of a truce, met with President Bashar al-Assad of Syria in Damascus as he sought to enlist Syria, a key backer of Hamas, in maneuvers toward a cease-fire. But while the French leader launched an impassioned plea for an end to the fighting, describing it as “unbearable,” Mr. Assad accused Israel of committing a war crime by invading Gaza and said it would pay “the highest price.”

The Israeli Army made no immediate comment on news reports, including Israeli radio, citing witnesses as saying that Israeli forces were now probing toward Khan Yunis in the south, a new development after concentrating their massive firepower on the north of Gaza.

The campaign has not proceeded without mishaps for Israel, which said on Tuesday that four of its soldiers in Gaza were killed by shells from their own tanks. The Israeli Army said three Israeli soldiers were killed by tank fire directed at a building they had occupied in northern Gaza, and a fourth soldier was killed in a separate incident, also apparently caused by a tank shell.

Casualties were reported mounting as the military confrontation broadened. Since launching its ground offensive, Israel has killed 130 Hamas fighters, Israeli officials say. Hamas has killed five Israelis by rocket fire and in combat. Palestinian medical officials estimated that the death toll during the 11-day war exceeded 560 on Tuesday and the U.N. said that about a quarter of those killed were civilians.

Defying Israeli and international demands, Hamas militants in Gaza fired more rockets into Israel Tuesday, one of them falling in Gadera.

The location was significant to many Israelis since Gadera, about 25 miles north of Gaza, is perceived as being linked to Tel Aviv, meaning that central Israel may now be vulnerable to Hamas rockets along with the southern cities that have borne the brunt of the missile fire. Shrapnel from the attack slightly injured a three-month-old baby, the army said.

Since the operation began, Israeli officials in Washington said, the number of rocket and mortar attacks from Gaza has fallen to about 20 a day from a peak of 80 on Christmas Day.

“The situation has obliged them to contract and pull back the rockets,” said Jeremy Issacharoff, the Israeli deputy chief of mission in Washington. “The rate of attrition is important,” he said, noting that Hamas was now launching fewer rockets than Israeli forces had expected.

In the north of Gaza, three Palestinian men were killed late Monday night when a United Nations Relief and Works Agency school compound was hit by Israeli fire, according to a statement released on Tuesday by the organization, which provides assistance to registered Palestinian refugees. More than 400 Palestinians from northern Gaza were taking refuge in the school in Gaza City at the time, and the building was clearly marked as a U.N. installation, the statement said.

The latest fighting coincided with a new and inconclusive diplomatic effort to bring pressure on Hamas to halt the rocket attacks — one of Israel’s conditions for a cease-fire, along with the destruction of Hamas as a fighting force and measures to prevent the Islamic militants from re-arming.

In Damascus, President Sarkozy met with President Assad after holding earlier talks in Egypt and discussions with Israeli leaders and Palestinian officials in the West Bank. Hamas is headquartered in Syria, and Mr. Assad is key ally of both Hamas and the Islamist Hezbollah movement in Lebanon. There was no immediate indication that the French leader had secured a commitment from Mr. Assad to put pressure on Hamas.

After talks with Mr. Sarkozy, Mr. Assad said that Israeli leaders “have not learned the lessons of the war in Lebanon” in 2006 when Hezbollah emerged politically strengthened from a bruising battle with Israel.

“Israel is falling into the same trap again and the Israelis will pay the highest price,” Mr. Assad said, calling the Israeli offensive a “war crime.”

Mr. Sarkozy said the violence “must stop immediately, as soon as possible.”

Both sides in the Gaza conflict have adopted uncompromising positions.

On Monday, the Israeli foreign minister, Tzipi Livni, said after a meeting with officials from the Czech Republic, Sweden and France that Israel would “change the equation” in the region. She added that in other conflicts, “countries send in forces in order to battle terrorism, but we are not asking the world to take part in the battle and send their forces in — we are only asking them to allow us to carry it out until we reach a point in which we decide our goals have been reached for this point.”

Israeli officials have said repeatedly that they are not ready to accept any cease-fire proposal that did not guarantee a permanent halt to rocket attacks as well as smuggling of weapons through tunnels under Gaza’s border with Egypt.

The Hamas leader Mahmoud Zahar, speaking from a hiding place in a recorded speech on Hamas television, said: “The Israeli enemy in its aggression has written its next chapter in the world, which will have no place for them. They shelled everyone in Gaza. They shelled children and hospitals and mosques, and in doing so, they gave us legitimacy to strike them in the same way.”

Israel said it had hit some civilian targets because they housed rockets, launchers or militants. It offered limited evidence of its claim.

Toward night on Monday, northern Gaza was the site of heavy fighting, including artillery, helicopter and tank fire, witnesses said. Plumes of smoke were visible in the night sky.

Inside Gaza City, windows are blown out, electricity is cut and drinking water scarce. While phones rang with the recorded threats against Hamas, leaflets dropped from airplanes littered the streets, saying: “Hamas is getting a taste of the power of the Israeli military after more than a week and we have other methods that are still harsher to deal with Hamas. They will prove very painful. For your safety, please evacuate your neighborhood.”

Israeli officials hope an eventual deal will be struck without engaging directly with Hamas, but Mark Regev, the spokesman for Prime Minister Ehud Olmert, said Israel would not exclude a tacit understanding.

“The endgame for us is threefold: that Hamas’s military machine would be substantially destroyed; two, Hamas understands that shooting rockets means paying a price they don’t want to pay; and three, there are mechanisms in place to prevent Hamas from rearming,” Mr. Regev said.

But as the offensive unfolds, so, too, evidence is mounting of a severe humanitarian crisis.

Maxwell Gaylard, United Nations humanitarian affairs coordinator, said at a Jerusalem news briefing on Monday that because of the attacks, people could not reach available food.

Children are hungry, cold, without electricity and running water, he said, “and above all, they’re terrified. That by any measure is a humanitarian crisis.”