Israeli forces and Palestinian groups say there have been intense battles in the northern Gazan town of Jabalia.
Israel's military said troops had killed "a large number of terrorists" in Jabalia's refugee camp, while Hamas said it had killed several soldiers.
Tens of thousands of civilians have fled Jabalia since Saturday, when the military said it would re-enter the area because Hamas had regrouped there.
Fierce fighting is also continuing around the southern city of Rafah.
The UN says almost 600,000 people taking refuge have been displaced since the start of an Israeli ground operation in the city's eastern outskirts nine days ago, including 150,000 people in the past 48 hours.
With tanks now pushing into built-up areas and the nearby border crossings closed or inaccessible, supplies and medical services are dwindling.
US Secretary of State Antony Blinken expressed deep concern about the impact of the Israeli operation, saying there was an urgent need to restore the full operational capacity of the two Rafah and Kerem Shalom crossings to make sure humanitarian assistance got into southern Gaza.
"What we don't want to see is a situation where we've basically reversed what's happened in recent months - where assistance was working its way through in the south but very little was getting to the north," he said.
But Israeli Prime Minister Benjamin Netanyahu pushed back against criticism of the military's operation, saying: "The humanitarian catastrophe that has been spoken of has not been realised, nor will it."
Israel launched a military campaign to destroy Hamas in response to the group's cross-border attack on southern Israel on 7 October, during which about 1,200 people were killed and 252 others were taken hostage.
More than 35,230 people have been killed in Gaza since then, according to the territory's Hamas-run health ministry.
The Israel Defense Forces (IDF) said in a statement on Wednesday that troops had begun "an operation against terrorist operatives and infrastructure in the centre of Jabalia camp" overnight.
"Over the past day, IDF troops in the area have engaged in intense battles with dozens of terrorist cells and eliminated a large number of terrorists," it added.
Hamas's military wing also reported battles inside Jabalia camp on Wednesday and said fighters had targeted Israeli troops with anti-tank missiles and various other explosive devices.
It claimed that a dozen soldiers had been killed in attacks on a bulldozer, a tank and a house in the Block 4 area of the camp. The IDF did not report any casualties.
The Hamas-affiliated Safa news agency reported that air and artillery strikes had caused widespread destruction in areas where the troops had advanced.
Meanwhile, the official Palestinian news agency, Wafa, said four people were killed in an overnight Israeli air strike on a house in Jabalia town.