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According to the UN aid agency Gaza is "being pushed into an abyss".
It is not clear at this stage just how many people are moving from the north of the Gaza Strip to the south, said the UN relief agency which operates inside the territory.
Earlier UNRWA's director of communications, Juliette Touma, told the BBC, "This is the worst we've ever seen. This is hitting rock bottom. This is Gaza being pushed into an abyss, there is tragedy unfolding as the world is watching. This is Gaza."
She added: "According to colleagues on the ground, there is an exodus. People are leaving. Those who can, with their cars, some are walking, some are carrying mattresses.
"People are terrified," she said. "Terrified."
The Israeli military is planning an attack by land, air and sea in Gaza Strip but gives no timings or specifics.
A ground offensive by Israel into Gaza is expected. Prime Minister Benjamin Netanyahu tells frontline soldiers: "The next stage is coming."
Netanyahu has warned 1.1m Palestinians living in north Gaza to evacuate south, and thousands have been fleeing by vehicle or on foot.
Meanwhile, the World Health Organization has condemned the evacuation order, and says forcing hospital patients to relocate would be "a death sentence".
As the clash enters its eighth day, more than 3,600 people have been killed on both sides.
More than 1,300 people were killed in Israel last weekend in an attack by Hamas, while more than 2,300 people have been killed in Israel's bombing campaign of the Gaza Strip launched in the aftermath.
In Egypt, an aid convoy has been waiting for hours to enter Gaza through the Rafah crossing.
Pictures show a long queue of lorries carrying supplies from Egypt and Turkey waiting at the city of Arish, near the crossing.
Several countries including the UK and the US, have urged their citizens in Gaza to move closer to the Rafah crossing so they can leave when the crossing reopens. They have warned any reopening would last only for a very limited time.
On Saturday US officials said they had been working to get hundreds of Palestinian-Americans to leave via Rafah, though it remains unclear whether they succeeded.
Bd-pratidin English/Golam Rosul
