During the day, the fighting appeared to move away from the northern end of the territory, towards more populous areas in the west, correspondents say.
Later, Israeli military sources and witnesses said Israeli tanks and heavy armour had taken up positions on either side of Gaza City, in effect cutting Gaza into two parts, from the Karni crossing to the Mediterranean Sea.
The town of Beit Hanoun was also reportedly surrounded.
Hamas officials and witnesses report major fighting in five areas: east of the Jabaliya refugee camp; in the Zeitoun area; near the site of the former Jewish settlement of Netzarim; in the centre of Gaza; and on the outskirts of Khan Younis.
Hamas said its fighters were in some cases engaged in "face-to-face battles" with Israeli soldiers.
Earlier, the Israeli military said the militants were not engaging its troops in close combat but using mortars and improvised bombs.
The Palestinian health ministry says 509 Palestinians, mostly civilians, have been killed since the Israelis began their assault on Gaza eight days ago.
It says 21 of the 70 people killed since the beginning of the ground offensive were children. Some 2,500 people have reportedly also been wounded.
The figures could not be independently verified. Israel is refusing to let international journalists into Gaza despite a ruling by its a supreme court to admit a limited number of reporters.
Hamas officials say that 10 of its fighters have so far been killed.
The Israeli military says one of its soldiers has been killed and 34 wounded in the ground offensive, three of them seriously. It believes about 80% of the Palestinians killed were Hamas members.
Battle to save injured
The BBC's Rushdi Abu Alouf in Gaza City says the fighting and Israeli positions have stopped desperately needed medical supplies getting through to hospitals that are struggling to cope with the casualties.
Many agencies say deliveries have been insufficient, and that it is difficult to get supplies to where they are most needed.
Sharon Lock, an activist with the International Solidarity Movement, a Palestinian-led organisation which campaigns against Israel's actions, told the BBC she had watched medical crews in Gaza working under terrible conditions:
"I went to one house where about four people had been injured," she said.
"One was completely killed, his legs were mangled and he'd lost a foot. There was a three-year-old child in the house that was injured, there were two men in their thirties who had severe internal injuries and we had to put them all in the one ambulance.
"We had to squash them in, you know, on top of each other, including the dead man, just to get them to some sort of place of safety."
An Israeli army spokesman, Capt Guy Spiegelman, told the BBC Hamas, not Israel, was to blame for causing civilians to suffer.
"If they just would stop firing those rockets, we wouldn't be in this situation," he said.
"If they would place their rockets - not amongst civilian population, but in open areas - then the civilians wouldn't need to suffer," the Israeli officer added.
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