The Israeli military says its forces have entered the centre of Khan Younis, the main city in southern Gaza.
It also said fierce clashes were taking place in Jabalia and Shejaiya in northern Gaza in the heaviest day of fighting so far.
Hamas said it had destroyed Israeli military vehicles and used snipers against troops.
The UN says the situation in southern Gaza is getting worse and distributing food there is nearly impossible.
Hundreds of people have been killed since Israel began bombing southern Gaza last week after a temporary ceasefire ended.
On Tuesday officials from the Hamas-run health ministry said at least 45 people had been killed in Israeli strikes on homes in the central city of Deir al-Balah.
Hamas officials say at least 16,248 people have been killed in Gaza since the start of the conflict, including more than 7,000 children and nearly 5,000 women.
Israel launched air strikes and a ground invasion of Gaza after Hamas attacked nearby military bases, kibbutzes and a music festival, killing about 1,200 people.
The Israeli military says it is working to destroy Hamas in Gaza and rescue 138 hostages it says are being held in Gaza.
However on Tuesday Israeli media reported that Israel’s Prime Minister, Benjamin Netanyahu, had told relatives of those still being held that there was “no possibility right now of bringing everyone home”.
Before a temporary truce collapsed last Friday, the Israeli ground operation was focused on the north of Gaza.
The Israel Defense Forces (IDF) issued a blanket order to all 1.1 million people living there to move south and many went to Khan Younis.
The Israeli government and military insists it is doing what it can to minimise civilian casualties in Gaza and blames Hamas for operating in densely populated areas.
After several days of telling people in some parts of Khan Younis to leave, the Israeli military on Tuesday told them instead to remain in designated shelters and hospitals.
“Don’t get out. Going out is dangerous. You have been warned,” said the leaflets, addressed to residents of six city districts.
It had previously told them to head towards Rafah near the border with Egypt and to an area near the coast called al-Mawasi.
Philippe Lazzarini, head of the UN agency for Palestinian refugees, said limits on the amount of aid allowed to enter Gaza from Egypt “strangles our operations”.
“The siege might become a major source of death,” he said.
The World Food Programme said the distribution of aid was now “almost impossible”.
On Tuesday Samah Ilwan, a mother-of-six, was waving empty bottles and pleading for help in the centre of Khan Younis. Her children were thirsty, she told the BBC.
“We have become like cats and dogs… We’re stranded in the street,” she said, adding: “I want to send a message to the world that we are innocent. We didn’t do anything wrong.”
