A UN peacekeeping official told the BBC News website the decision to move non-essential UN staff temporarily to locations out of the country - many to Entebbe in Uganda - had been taken after that incident, and as a prudent measure in anticipation of possible Sudanese reaction to the prosecutor's announcement that he is seeking a warrant against Mr Bashir.
In the US, the White House said President George W Bush was "gravely concerned" by increased insecurity in Darfur and the impact it is having civilians and aid workers.
His spokeswoman said that the government of Sudan needed to live up to its commitment to provide increased security to humanitarian envoys.
As of May this year, the joint UN-African Union Darfur mission, Unamid, included nearly 9,600 uniformed personnel and about 1,300 civilian staff, both international and local.
It is not clear how many will be withdrawn. But Gen Martin Luther Agwai, Unamid force commander, said the peacekeepers would maintain their unit strength and would not stop patrolling.
Sudan's ambassador to the UN rejects the genocide charges
"We will continue to protect the UN personnel and UN facilities that are here and we will continue to help the humanitarian organisations to continue to do their job of rendering humanitarian services to the people in Darfur," Gen Agwai said.
A Sudanese official told the BBC that he had been informed by Unamid the evacuation would begin on Tuesday.
"This is a unilateral decision which the Sudanese government was not involved in," Mr Mutrif Seddeek told the BBC.
While clarifying that the ICC is an independent body over which it has no influence, the UN is nevertheless bracing itself for possible increased difficulties in western Sudan, the BBC's David Bamford reports.
But it is continuing to run, alongside other NGOs, large-scale humanitarian operations there and has thousands of peacekeepers in place, our correspondent notes.
Security Council attempts
Sudan's president was accused of genocide, crimes against humanity and war crimes in Darfur by ICC prosecutor Luis Moreno-Ocampo.
ACCUSATIONS AGAINST BASHIR
Killing members of the Fur, Masalit and Zaghawa ethnic groups
Causing these groups serious bodily or mental harm
Inflicting conditions of life calculated to bring about these groups' physical destruction
He told judges at The Hague that Mr Bashir bore criminal responsibility for alleged atrocities committed over the past five years.
Sudan has refused to hand over two suspects who Mr Moreno-Ocampo charged last year, Humanitarian Affairs Minister Ahmad Harun and militia leader Ali Kushayb.
It has also labelled Mr Moreno-Ocampo a criminal, and warned that any indictment could stall peace talks and cause mayhem in Sudan.
Sudanese cabinet minister al-Samani al-Wasila told the BBC the accusations were "politically motivated".
The BBC's Laura Trevelyan at the United Nations in New York says that Mr Ocampo was asked by the UN Security Council to investigate the killings in Darfur - and this means the 15-member body also has the power to suspend his work on the case for a year.
Behind the scenes, our correspondent says, Sudan has been lobbying China, Russia and African nations on the Security Council to see if it can win the nine votes needed to pass a resolution to do that.
It would be difficult for countries which recognise the ICC, like the Europeans, to vote to stop the prosecutor's work, as this would seem to undermine the court, she says.
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