President George W Bush has refused to set a timetable to hand over ex-President Saddam Hussein to the interim Iraqi government.
Saddam was seized eight months after his regime fell
Mr Bush told reporters "appropriate security" must be in place first.
The US and the Iraqi government do not want "lax security and for Saddam Hussein not to stand trial", Mr Bush said outside the White House.
Iraq's prime minister has said he wants Saddam Hussein to be handed over before the 30 June transfer of power.
Iyad Allawi told al-Jazeera television on Monday this would happen "within the next two weeks".
The Red Cross has said he must be freed or charged before the end of June.
The head of Iraq's special tribunal on war crimes, Salem Chalabi, said the country would "very shortly" have the proper facilities to detain Saddam Hussein.
Mr Chalabi said the Iraqi authorities were also preparing to issue arrest warrants for the country's former rulers.
The US-led coalition would hand them over "provided we show arrest warrants based on reasonable grounds", he told the French news agency, AFP.
The Red Cross insisted on Sunday that the former Iraqi leader, designated a POW, cannot continue to be held by US forces after the conflict ends.
"His case is the same as all other prisoners of war," said Nadia Doumani, who speaks for the International Committee of the Red Cross in Baghdad.
The US expects to hold thousands of Iraqis after 30 June
Geneva Conventions state that if POWs are only being held because they are enemy combatants - not charged with any crime - they have to be released when the conflict ends.
"Saddam Hussein can be convicted for war crimes, for crimes against humanity... then he can be tried and prosecuted," said Ms Doumani.
"If he is not charged, then the law says that at the end of war, of occupation, he should be released," she added.
Saddam Hussein has been held at an undisclosed location since his capture by US forces in December and is being interrogated by the CIA and FBI.
He is expected to face charges of genocide and crimes against humanity.
The ICRC has paid two humanitarian visits to check the conditions under which he is being held.
The US army has said it will transfer up to 1,400 prisoners to Iraqi custody before 30 June - or release them entirely.
However, it plans to continue holding between 4,000 and 5,000 prisoners whom it has deemed a threat to the coalition.
About 400 prisoners were freed from Abu Ghraib on Monday and dozens of others have been informed of their impending release.