"Make the deal in Gaza, get the hostages back," Trump posted on his Truth Social platform early on Sunday.
Israeli Prime Minister Benjamin Netanyahu was due to hold talks later in the day on the progress of Israel's offensive.
A senior security official said the military will tell him the campaign is close to reaching its objectives, and warn that expanding fighting to new areas in Gaza may endanger the remaining Israeli hostages.
But in a statement posted on X and text messages sent to many residents, the military urged people in northern parts of the enclave to head south towards the Al-Mawasi area in Khan Younis, which Israel designated as a humanitarian area. Palestinian and UN officials say nowhere in Gaza is safe.
"The (Israeli) Defense Forces is operating with extreme force in these areas, and these military operations will escalate, intensify, and extend westward to the city center to destroy the capabilities of terrorist organizations," the military said.
The evacuation order covered the Jabalia area and most Gaza City districts. Medics and residents said the Israeli army's bombardments escalated in the early hours in Jabalia, destroying several houses and killing at least six people.
In Khan Younis in the south, five people were killed in an airstrike on a tent encampment near Mawasi, medics said.
The escalation comes as Arab mediators, Egypt and Qatar, backed by the United States, begin a new ceasefire effort to halt the 20-month-old conflict and secure the release of Israeli and foreign hostages still being held by Hamas.
Interest in resolving the Gaza conflict has heightened following US and Israeli bombings of Iran's nuclear facilities.
A Hamas official told Reuters the group had informed the mediators it was ready to resume ceasefire talks, but reaffirmed the group's outstanding demands that any deal must end the war and secure an Israeli withdrawal from the coastal territory.
Hamas has said it is willing to free remaining hostages in Gaza, 20 of whom are believed to still be alive, only in a deal that will end the war.
Israel says it can only end it if Hamas is disarmed and dismantled. Hamas refuses to lay down its arms.
The latest bloodshed in the decades-old Israeli-Palestinian conflict was triggered in October 2023 when Hamas attacked Israel, killing 1200 people and taking 251 hostages, Israeli tallies show.
Israel's subsequent military assault has killed more than 56,000 Palestinians, according to the Gaza health ministry, displaced almost the entire 2.3 million population, plunged the enclave into a humanitarian crisis and left much of it in ruins.
