The military evacuation demand, which could signal an imminent attack on neighbourhoods in Deir al-Balah, alarmed the families of Israeli hostages, who fear their relatives are being held there.
Much of Gaza has been reduced to a wasteland during more than 21 months of war and there are fears of accelerating starvation.
Health officials at Al Shifa Hospital in northern Gaza said at least 30 Palestinians were killed and dozens of others wounded by Israeli fire as crowds gathered to await the entry of United Nations aid trucks.
Israel's military said it was checking the report.
Palestinian health officials said hundreds of people could soon die as hospitals were inundated with patients suffering from dizziness and exhaustion due to the scarcity of food and a collapse in aid deliveries.
"We warn that hundreds of people whose bodies have wasted away are at risk of imminent death due to hunger," the health ministry, which is controlled by Hamas, said.
The UN also said civilians were starving and needed an urgent influx of aid.
The Israeli military dropped leaflets from the sky ordering people in several districts in southwest Deir al-Balah, where hundreds of thousands of displaced Gazans have been sheltering, to leave their homes and head further south.
"The (Israeli) Defense Forces continues to operate with great force to destroy the enemy's capabilities and terrorist infrastructure in the area," the military said, adding that it had not entered these districts during the current conflict.
Israeli sources have said the reason the army has so far stayed out is it suspects Hamas might be holding hostages there.
At least 20 of the remaining 50 hostages in captivity in Gaza are believed to be alive.
Hostage families demanded an explanation from the army.
"Can anyone (promise) to us that this decision will not come at the cost of losing our loved ones?" the families said in a statement.
Some Palestinians suggested the move on Deir al-Balah might be an attempt to put pressure on Hamas to make more concessions in long-running ceasefire negotiations.
Israel and Hamas are engaged in indirect talks in Doha aimed at reaching a 60-day truce and hostage deal, although there has been no sign of a breakthrough.
The war began when Hamas-led militants stormed into Israel on October 7, 2023, killing 1200 people and taking 251 hostages back to Gaza.
The Israeli military campaign against Hamas in Gaza has since killed more than 58,000 Palestinians according to health officials, displaced almost the entire population and plunged the enclave into a humanitarian crisis.
Local health authorities said a total of 45 people had been killed in separate Israeli gunfire and air strikes across Gaza on Sunday.
Residents said it was becoming impossible to find essential food such as flour.
The Gaza health ministry said at least 71 children had died of malnutrition during the war, and 60,000 others were suffering from symptoms of malnutrition.
Food prices have increased beyond what most of the population of more than two million can afford.
Several people who spoke to Reuters via chat apps said they either had one meal or no meal in the past 24 hours.
"As a father, I wake up in the early morning to look for food, for even a loaf of bread for my five children, but all in vain," said Ziad, a nurse.
UNRWA, the UN refugee agency dedicated to Palestinians, demanded Israel allow more aid trucks into Gaza, saying it had enough food for the entire population for more than three months that was not being allowed in.
Israel has denied accusations it is preventing aid from reaching Gaza and has accused Hamas of stealing food, an allegation Hamas denies.
