About 1,500 bodies of Hamas militants were found in Israeli territory, military officials have said.
Israeli spokesperson Richard Hecht said the military had largely gained control in the country’s south and “restored full control” over the border on the fourth day of fighting following an unprecedented surprise attack.
No Hamas fighters have crossed into Israel since Monday night, although infiltrations could still be possible, Mr Hecht added.
Israel has previously reported 900 soldiers and civilians killed, and Palestinian authorities have reported about 700 deaths in Gaza and the West Bank.
The report from the UN Office for the Co-ordination of Humanitarian Affairs (OCHA) comes amid ongoing Israeli airstrikes and the military wing of the militant Hamas group saying it will kill one Israeli civilian captive any time civilians are targeted in their homes in Gaza “without prior warning”.
UNRWA, the UN agency for Palestinian refugees, said it is hosting more than 137,000 people in schools across the territory.
The UN report said airstrikes have razed 790 housing units and severely damaged 5,330 in Gaza, with damage to water, sanitation and hygiene facilities disrupting service for more than 400,000 people.
Catherine Russell, executive director of UN children’s agency Unicef, called for the “immediate and safe release of any children being held hostage in Gaza”.
She said: “With the humanitarian situation rapidly deteriorating, humanitarian actors must be able to safely access children and their families with lifesaving services and supplies – wherever they may be.
Israel’s military said a deputy Israeli commander identified as Alim Abdallah was among the dead after being killed in clashes on the northern border with Lebanon.
Lebanon’s Hezbollah militant group said five of its members were killed by shelling into southern Lebanon and it retaliated with a volley of rockets and mortars at two Israeli army bases across the border.
The Israeli military also said it struck two tunnels used by Hamas militants to enter Israeli territory.
An airstrike in Gaza City also killed two Palestinian journalists early on Tuesday, according to the Palestinian news agency Wafa, who identified them as editor Saeed Al-Taweel and photographer Mohammed Sobih.
Israel increased airstrikes on the Gaza Strip on Monday as it sealed the territory off from food, fuel and other supplies.
In a video statement on Monday, Israel’s foreign minister warned Hamas against harming any of the hostages who were taken from Israel and are being held in Gaza.
Eli Cohen said Israel was committed to bringing the hostages home “in the spirit of mutual responsibility”.
He said: “We demand Hamas not to harm any of the hostages. This war crime will not be forgiven.”
“We have only started striking Hamas,” Israeli Prime Minister Benjamin Netanyahu said in a nationally televised address. “What we will do to our enemies in the coming days will reverberate with them for generations.”
Hamas and other militants in Gaza say they are holding more than 130 soldiers and civilians snatched from inside Israel.