Prime Minister Benjamin Netanyahu said Israel is likely to maintain control of security in Gaza once Hamas is defeated.
The move into Gaza City risks a further escalation in casualties, while Mr Netanyahu's comments pointed to the uncertainty surrounding the end game of a war that Israel says will go on for some time until it destroys Hamas rule.
Memorial events are planned in Tel Aviv and Jerusalem to mark the 30th day - a milestone in Jewish mourning - since the deaths of around 1,400 in Hamas's October 7 rampage in southern Israel which sparked the war.
About 240 people abducted by Hamas during the attack remain in the militants' hands. More than 250,000 people have evacuated homes near Gaza and the Lebanon border amid continuous militant rocket fire toward Israeli cities and towns.
A month of relentless bombardment in the Gaza Strip has killed more than 10,300 Palestinians, two-thirds of them women and minors, according the Health Ministry of the Hamas-run territory. More than 2,300 are believed to be buried from strikes that reduced entire city blocks to rubble.
Around 70% of Gaza's 2.3 million people have fled their homes, and more than 700,000 are crowded into UN schools-turned-shelters, relying on a trickle of aid and their own daily foraging for food and water from supplies choked off by weeks of siege.
Israeli ground troops have battled Palestinian militants inside the Gaza Strip for over a week, cutting the territory in half and encircling Gaza City.
Chief military spokesman Rear Admiral Daniel Hagari said Israeli ground forces "are located right now in a ground operation in the depths of Gaza City and putting great pressure on Hamas".
On Monday, Mr Netanyahu said the military had killed several thousand Hamas fighters since the war began. The Gaza Health Ministry's death toll does not distinguish between civilians and combatants - and killed fighters not taken to hospital would not be in its count.
Israel unleashed another wave of strikes across the Gaza Strip on Tuesday as hundreds more Palestinians fled Gaza City to the south.
Hundreds of thousands have heeded Israeli orders to head to the southern part of Gaza, out of the ground assault's path. Others are afraid to do so since Israeli troops control part of the north-south route.
But bombardment of the south has also continued. An Israeli air strike destroyed several homes early on Tuesday in Khan Younis. An Associated Press journalist at the scene saw first responders pulling five bodies - including three children - from the rubble.
In the town of Deir al-Balah, rescue workers brought out at least four dead and a number of wounded children from the wreckage of a flattened building, witnesses said.
Israel says it targets Hamas fighters and infrastructure and accuses the group of endangering civilians by operating among them.
The Israelis have vowed to remove Hamas from power and crush its military capabilities - but neither Israel nor its main ally, the US, has said what would come next.
Mr Netanyahu said Gaza should be governed by "those who don't want to continue the way of Hamas", without elaborating.
"I think Israel will, for an indefinite period, have the overall security responsibility because we've seen what happens when we don't have it. When we don't have that security responsibility, what we have is the eruption of Hamas terror on a scale that we couldn't imagine," he said.
He did not make clear what shape that security control would take. US officials have advised that Israel should not re-occupy the Gaza Strip.
Israeli officials say the offensive against Hamas will last for some time and acknowledge they have not yet formulated a plan for afterwards.
The defence minister has said Israel does not seek a long-term reoccupation of Gaza but predicted a lengthy phase of low-intensity fighting against "pockets of resistance". Other officials have spoken about establishing a buffer zone that would keep Palestinians away from the Israeli border.
Israel withdrew troops and settlers in 2005 but kept control over Gaza's air space, coastline, population registry and border crossings, excepting one into Egypt.
Hamas seized power from forces loyal to President Mahmoud Abbas in 2007, confining his Palestinian Authority to parts of the occupied West Bank. Since then, Israel and Egypt have imposed a blockade on Gaza to varying degrees.