Residents said Israeli tanks and heavy bulldozers broke the border fence to gain access to the Gaza Strip.
An Israeli military spokesman confirmed that "an army operation is under way".
Six Hamas men were killed in the clashes, three by missiles and two by gunfire, Abu Obeida, a spokesman for Hamas's military wing, said.
One Hamas fighter died later in hospital due to his wounds.
The Islamic Jihad faction said one of its members also died in the fighting.
Suheir Naim, Hani's widow, said feelings of hatred were increasing.
"What's the fault of my children and other children who lose their fathers? When a small Israeli child is killed, the whole world turns upside down. But who cares about us?" she said.
David Chater, Al Jazeera's correspondent in Gaza, said the "war in Gaza" was escalating on two fronts.
"On the economics front the Israeli high court has authorised further cuts on electricity and fuel supplies into Gaza," he reported.
The cuts to electricity were permitted after Israel's supreme court rejected a petition by 10 Israeli and Palestinian human rights organisations last week, challenging Israel's planned reductions.
"And on the military front," he said. "Israel is increasing its attacks on the armed wing of Hamas.
"But civilians are falling on both fronts."
Bassim Naim, the former government health minister, told Al Jazeera that attempts by the Israelis to break the Palestinians commitment would not work.
"As citizens under occupation, it is our normal right to use all the legal ways of resistance, to liberate our lands," he said.
Attacks by Israel on Gaza have killed 19 Palestinians, mostly fighters, in the past week alone.
Israeli regularly carries out raids on the Gaza Strip in what it says is an effort to stop Palestinian fighters from launching homemade rockets into southern Israel.
On Wednesday, a Hamas rocket hit Kibbutz Beeri, an Israeli communal village about 6km from the border fence, police said.
Two sisters, aged 12 and two, were hurt as they played in their yard, but escaped serious injury.
Their mother was taken to a hospital and treated for shock.
On Thursday, Ehud Barak, the Israeli defence minister, said he would continue the bombardment of Gaza as long as rocket fire continued.
"If the rocket fire from Gaza continues, we will intensify our operations and strikes against the other side, until a solution is found," he said.
Al Jazeera's correspondent said there was no answer, as far as the Israel security forces were concerned, to these rocket attacks from the Gaza Strip.
|Israel's defence minister said raids on Gaza |
would continue as long as rocket fire did [AFP]
"That's why many people fear that the Israeli ministry of defence is beginning to look at plans for a much larger incursion into the Gaza Strip," Chater reported.
On Thursday, Nabil Abu Rudeineh, a media adviser to Mahmoud Abbas, the Palestinian president, said that Abbas was trying to co-ordinate a ceasefire with Israel in order to stop the violence in the Gaza Strip, despite the territory being controlled by Hamas.
"Just as the president expressed his readiness to lift the suffering of the Palestinian people in Gaza by re-assuming responsibility for the border crossings, the president is also ready to try to reach a mutual ceasefire [with Israel] in order to stop the ongoing massacres in the Gaza Strip," he said.
Hamas promptly rejected the offer. Spokesman Fawzi Barhoum branded it a "blackmail attempt against the Palestinian people whom [Abbas] has left to be massacred".
Following a meeting with Abbas, Tony Blair, the Middle East Quartet envoy, said on Thursday that efforts have to continue to help the population of Gaza while "isolating the extremists" until national reconciliation "between the West Bank and Gaza" is achieved.
"There will be one Palestinian state, not two," he said.