Iraqi leaders say they want Saddam Hussein put on trial in Iraq for crimes against humanity following his capture.
The US said Saddam Hussein offered no resistance
The former president is in US custody after being found in a hole under a mud hut near his home town of Tikrit.
The US promised he would "face justice" but gave no details while the head of Iraq's interim government said he wanted him "judged by Iraqi judges".
Amid celebrations for his capture, two Iraqi police stations were attacked, reportedly killing about seven people.
The former president has been shown on television being examined by an American military doctor at an undisclosed location after his capture.
US Defence Secretary Donald Rumsfeld said he was being accorded the privileges of being a prisoner-of-war under the Geneva Convention although America was "not defining him as such".
As for the former president's fate, Mr Rumsfeld said it would be decided at a senior level.
"No-one would want to turn anyone over until and unless there was a process in place that was acceptable and appropriate and would ensure that he would be brought to justice," he said.
He said the prisoner was not co-operating with his interrogators.
The current head of the US-appointed Iraqi Governing Council (IGC), Abdel Aziz al-Hakim, said Saddam Hussein would be judged by Iraqi judges at a tribunal working and located in Iraq "under the supervision of international experts".
Saddam Hussein was found in a hole in the ground
Adnan Pachachi, another IGC member who saw Saddam Hussein on Sunday when he was brought to Baghdad for further identification, said the ex-president had appeared "unrepentant and even defiant", insisting he had been a "just and firm ruler".
US President George W Bush said the capture had brought to an end a "dark and painful" era for the Iraqi people but he warned it did not mean the end of violence in Iraq.
A special tribunal was set up in Iraq last week to try leading members of the former government.
Some human rights groups say an international tribunal - without the power to award the death penalty - would be preferable to a trial in Iraq.
The news of Saddam Hussein's arrest prompted scenes of jubilation in many parts of Iraq and was welcomed by world leaders including some who had opposed the war to oust him.
President Chirac of France said it would strongly contribute to democracy and stability in Iraq.
Kuwait - invaded by Saddam Hussein's forces 13 years ago - called it the moment the whole world had been waiting for.
In some Arab countries, there was amazement and disappointment that the Iraqi leader had given in to his pursuers without a fight, but also continuing admiration.
The US said the operation to capture the former president involved various US units and was launched within 24 hours of receiving a tip-off from a member of Saddam Hussein's clan.
Military officials described the sequence of events:
About 600 US troops descend on al-Dawr, south of Tikrit, and conduct intensive searches
US forces find a "rural farmhouse" and cordon off an area of about 2km by 2km
- "Spider hole" or cellar located next to a hut - the narrow hole covered with a rug, bricks and dirt and about six to eight feet (1.8m to 2.5m) deep, with a styrofoam insert
- Saddam Hussein found inside "like a rat", armed with a pistol, and arrested at 2030 local time (1730 GMT) on Saturday - US says he offers no resistance
- Two unidentified people said to be "close allies" of Saddam Hussein arrested and weapons and more than $750,000 cash confiscated.
US officials have made clear that the former Iraqi leader faces intensive interrogation both over what he knows about the ongoing insurgency against the US-led occupation and about alleged weapons of mass destruction - the basis for going to war.
Anonymous US officials who have reviewed transcripts of Saddam Hussein's initial questioning are quoted as saying it is full of rhetoric.
Time magazine reported that he talked about the Iraqi people being in bondage but that he has denied possessing weapons of mass destruction.