"Now is a time for war," Defence Minister Yoav Gallant said on Thursday as Israeli warplanes continued pounding Gaza in retaliation for the weekend attacks by Hamas militants that killed more than 1,300 Israelis, mostly civilians.
The Israeli military said it would operate significantly in Gaza City in the coming days and civilians would only be able to return when another announcement was made.
"Civilians of Gaza City, evacuate south for your own safety and the safety of your families and distance yourself from Hamas terrorists who are using you as human shields," the military said in a statement.
"Hamas terrorists are hiding in Gaza City inside tunnels underneath houses and inside buildings populated with innocent Gazan civilians."
A Hamas official said the Gaza relocation warning was "fake propaganda" and urged citizens not to fall for it.
The United Nations said it considered it impossible for such a movement of people to take place "without devastating humanitarian consequences."
Israel's ambassador to the UN, Gilad Erdan described the UN's response to Israel's early warning to the residents of Gaza as shameful.
Israel has vowed to annihilate the Hamas militant group which led the attacks on Saturday.
The Israeli military said in a statement on Friday that it struck 750 military targets in northern Gaza overnight, including what it said were Hamas tunnels, military compounds, residences of senior operatives and weapons storage warehouses.
However, a ground invasion of Gaza poses serious risk with Hamas holding scores of hostages kidnapped in the assault.
The Gaza Strip, home to 2.3 million people, is under siege by Israel, which has pounded Hamas targets in the enclave and killed more than 1,500 Palestinians in retaliatory attacks since the weekend incursions.
The International Committee of the Red Cross (ICRC) said fuel powering emergency generators at hospitals in Gaza could run out within hours and the United Nations World Food Program (WFP) warned food and fresh water were running dangerously low.
"The human misery caused by this escalation is abhorrent, and I implore the sides to reduce the suffering of civilians," ICRC regional director Fabrizio Carboni said.
The United Nations Palestinian refugee agency (UNRWA) said it had relocated its central operations centre and international staff to Gaza's south.
Seeking to build support for its response, Israel's government showed US Secretary of State Antony Blinken and NATO defence ministers graphic images of children and civilians they said Hamas had killed in a weekend rampage in Israel.
