Gaza/Tel Aviv - Israel outlined conditions for a truce in Gaza Tuesday, rebuffing an increasingly vocal international push for an immediate end to the fighting, including an attempt by Arab states to pass a draft resolution at the UN Security Council.

The Gaza offensive will continue until Hamas completely stops is rocket attacks against Israel, and a mechanism is in place to guarantee that the rocket fire will not resume and that no new weapons are smuggled into Gaza in the future, Prime Minister Ehud Olmert said.

Israel will not settle for a temporary solution, he told an EU delegation - in Jerusalem to push for an immediate truce or at least a humanitarian respite that would allow international organizations to safely distribute food parcels in Gaza.

At least 35 Palestinians were killed since midnight, raising the total Palestinian toll to at least 586 killed and 2,600 wounded since the Israeli offensive got underway Saturday 27. At least one-fourth of the dead are civilians, according to UN staff on the ground.

At least 12 members of the same family, al-Daia, were killed when Israel shelled or rocketed their house in southern Gaza City's Zaytoun neighbourhood, hospital officials said.

The neighbourhood is one of the areas on whose outskirts Israeli troops have taken up positions.

The troops, who entered Gaza on Saturday after a week of relentless airstrikes, have encircled Gaza City - which faces the Mediterranean on the west - from the south, east and north, and have also taken up positions on the eastern outskirts of Jabaliya, a crowded refugee camp to the north and considered the hub of Palestinian rocket-launching squads, and of Khan Younis in the south.

A Palestinian militant leader however denied reports that the ground troops had advanced into populated areas, saying they were still occupying largely open areas.

Heavy fighting took place in those areas, with Qassam Brigades saying in a statement sent to journalists that its militants were confronting the Israeli soldiers with anti-tank missiles, mortar shells, roadside bombs, and "face to face" exchanges of fire.

Residents said the militants were launching their anti-tank missiles and mortar shells at the Israeli troops from behind and from the roofs of civilian homes. When Israeli fighter jets or helicopter gunships then locate the source of the fire and strike back, the militants have run away and civilians are killed in their stead, they said.

Three civilians were also killed when a school made into a shelter by a UN agency (UNWRA) in western Gaza City was hit by an Israeli naval shell, a spokeswoman said.

Five Israeli soldiers have been killed since the ground fighting started Saturday, three of them late Monday when an Israeli tank erroneously shelled a house on the outskirts of Jabaliya, in which the troops had taken cover from Palestinian militants firing mortar shells. Another 24 Israeli soldiers were injured. Another Israeli soldier, a paratrooper officer, was killed late Monday in another friendly-fire incident, also in northern Gaza.

Hamas' armed wing, the Qassam Brigades, said in a statement that it had set a "trap" for the soldiers.

An Israeli military spokesman said the purpose of the ground invasion was to arrest and question Hamas fighters, find and destroy Hamas "infrastructure" - rocket storage sites - and hold territory from where militants have been launching rockets into Israel.

Soldiers have already detained and questioned dozens of suspected militants, the military spokesman spokesman said, saying some of them were taken to Israel and some were questioned inside Gaza.

Hamas nevertheless has for the time being succeeded in launching more rockets into Israel, firing more than 40 Monday and at least another 27 Tuesday by the late afternoon. The Israeli town of Gedera, located north-east of the port city of Ashdod and over 40 kilometres from the most northern tip of Gaza, was hit for the first time, where a three-month-old baby was slightly injured.

As it continued its ground offensive, Israel also kept up its airstrikes, attacking another 50 targets Tuesday, the military said, including more smuggling tunnels, rocket storage depots and the Saraya security headquarters in downtown Gaza City, already partially destroyed on the second day of the offensive Sunday last week.

Olmert, meeting French President Nicolas Sarkozy late Monday, meanwhile urged the Security Council, presided over by France this month and scheduled to convene at 2200 GMT on Tuesday, to give Israel "freedom to act" against Hamas and other militants firing rockets from Gaza.

Palestinian President Mahmoud Abbas was to address the council, which is meeting for the second time since Saturday.

Sarkozy, in Damascus later Tuesday, urged Syrian President Bashar al-Assad to exert pressure on Hamas to end its rocket attacks. He called on Syria to "convince Hamas to choose the voice of reason and the path of peace and reconciliation."

"I know Syria's influence on a number of parties," he said. Damascus hosts Hamas' foreign headquarters and leader-in-exile Khaled Meshaal. But Assad held Israel responsible, calling its assault on Gaza "barbarian" and a "war crime" and saying it must stop immediately.

While EU External Relations Commissioner Benita Ferrero-Waldner voiced harsh words in Jerusalem Tuesday, demanding at least a humanitarian lull in the Gaza fighting, US President George W Bush on Monday backed Israel and said "All of us, of course, would like to see ... violence stop, but not at the expense of an agreement that does not prevent the crisis from happening again."