But information about what is happening is limited as Israel has barred foreign reporters from entering Gaza.
An Israeli army spokesman said three soldiers were killed and three were severely wounded when a tank shell hit a "structure" where the soldiers were located.
"Twenty additional soldiers were lightly to moderately wounded in the same incident," he added.
After night fell on Monday, Israeli troops were reported to be battling Palestinian militants on the outskirts of the northern Shujaiya district of Gaza City.
Hamas and Islamic Jihad said their fighters had engaged Israeli soldiers with machine guns and rockets. Residents reported hearing loud explosions and heavy gunfire.
One report said Israeli forces had to use artillery and helicopter gunships to drive back the Palestinian fighters.
Elsewhere, Israeli tanks were moving towards the Bureij refugee camp in central Gaza, according to the Reuters news agency. Fighting was earlier witnessed around Beit Hanoun and the Jabaliya refugee camp.
An Israeli military spokesman said the assault was going according to plan
The BBC's Hamada Abu Qammar in Gaza City said residents on the outskirts were heading into the centre, seeking safe places away from the fighting.
Israeli Defence Minister Ehud Barak said Hamas had suffered a "hard blow", but insisted the offensive would continue.
"We still haven't reached our objectives," he told Israeli MPs.
A senior Hamas leader in the Gaza Strip, Mahmoud Zahhar, told Palestinians in a televised address that they were headed for "victory" against Israel.
Israelis "have legitimised the murder of their own children by killing the children of Palestine", he said.
The Israeli army said Palestinian militants had fired 20 missiles into southern Israel on Monday.
For the people of Gaza, living conditions are deteriorating sharply. Supplies of fuel, food, water, and wheat are said to be running desperately low.
Middle East envoy Tony Blair: 'We want an immediate ceasefire'
A spokesman for Unrwa, the UN aid agency for the Palestinians, said food was urgently needed and people were facing "serious hunger", with supplies for just 48 hours.
"One million people are without electricity. Crucially the hospitals in Gaza are running on emergency generators. This in my book amounts to a humanitarian crisis," Christopher Gunness told the BBC.
Israel said it had allowed a convoy of 80 lorries carrying food and medicines through Gaza's southern frontier.
Away from the front line, diplomatic efforts to halt the fighting in the Gaza Strip have been moving into high gear.
French President Nicolas Sarkozy held talks with Palestinian leaders in the West Bank town of Ramallah at the start of a series of meetings across the Middle East.
In remarks broadcast on French radio, he "condemned" the Israeli ground offensive, but said Hamas had also "acted in an irresponsible and unforgivable manner" by ending a six-month ceasefire with Israel and resuming firing rockets.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.