Civilians are fleeing northern Gaza by car, on the back of trucks and on foot after an Israeli warning that civilians should move south.
About 1.1 million people living in northern areas have been told to leave in the next day.
The Israeli military said it knew it would take longer than that to move everyone but blamed Hamas for telling people to ignore the call.
The UN called on Israel to withdraw its order, warning of “devastating humanitarian consequences”.
Israel has massed soldiers near Gaza ahead of an expected ground offensive into the densely populated enclave.
Hamas fighters kidnapped at least 150 people and took them into Gaza during brutal attacks on Israel at the weekend that killed 1,300 people.
More than 1,500 people have been killed in Gaza since Israel launched retaliatory air strikes, which continue.
A total blockade is being enforced with fuel, food and water running out. Israel says it won’t lift the restrictions unless Hamas frees all hostages