The reported move by Israeli forces into Khan Younis followed heavy fighting overnight around Gaza City and on the edges of the Deir al-Balah and Bureij refugee camps in central Gaza.
Hamas and Islamic Jihad said their fighters had engaged Israeli soldiers with machine-guns and rockets in Gaza City. Residents reported hearing loud explosions and heavy gunfire.
The Israeli army said three of its soldiers were killed by a shell from an Israeli tank which hit their position.
A fourth Israeli soldier was killed in fighting in the north of the Gaza Strip, military officials said.
Information about what is happening inside Gaza is limited as Israel has barred foreign reporters from entering.
The escalation in fighting comes as international pressure for a ceasefire is growing.
Palestinian President Mahmoud Abbas is due to attend a UN Security Council meeting on Tuesday, along with the foreign ministers of Saudi Arabia, Egypt, France and Britain, in an attempt to put pressure on the Security Council to act decisively.
French President Nicolas Sarkozy is in the Middle East trying to gather support for a ceasefire and Czech Foreign Minister Karel Schwarzenberg, whose country holds the EU presidency, is also in the region.
Israeli officials have so far rejected calls for a truce.
A spokesman for Prime Minister Ehud Olmert said any ceasefire would have to include provisions to prevent Hamas from re-arming in order to prevent future rocket attacks on Israel.
Hamas has said that Israeli attacks on Gaza must stop and the crossings into the territory, which Israel controls, must be fully opened.
Living conditions in Gaza are reported to be deteriorating sharply, with supplies of fuel and food not getting to where they are needed. Many in Gaza are also without running water or electricity.
Fears are growing over mounting civilian casualties in Gaza
The UN says a million people in the territory are without electricity and many are facing "serious hunger" within days.
Thousands of Gazans are reported to have fled their homes, despite the dangers of moving around outdoors in the territory. Many others are staying in their homes, unable to venture outside to look for supplies due to the fighting.
Tamer, in Beit Lahiya, told the BBC that Israeli tanks were just outside his home, firing.
"We are all in shock, very scared. We haven't slept for days. We are all in one room on the ground floor - it's the only room in the house which has glass in the windows."
Norwegian doctor Mads Gilbert, one of two foreign doctors working at Gaza's biggest hospital, al-Shifa, said operating rooms were full and people were dying because of a lack of supplies.
Dominic Nutt, of the aid agency Save The Children, told the BBC that workers in the territory were reporting rapidly deteriorating conditions:
"They don't have any water most of the day, there is no electricity, they are freezing cold, the windows have to be left open to stop them smashing when the bombs fall.
"Children are at risk from hypothermia, they are malnourished, there is not enough food, the situation is getting desperate."
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