US forces are training their sights on Bin Laden now
A news agency close to the Taleban, the Afghan Islamic Press (AIP), quoted a member of Kandahar's new tribal council as saying Mullah Omar had fled from the city.
But an American official said Mullah Omar was still believed to be in Kandahar despite the report - and would soon be caught.
Hamid Karzai, head of Afghanistan's new United Nations-backed administration, said Mullah Omar would be arrested if found.
"I have given him every chance to denounce terrorism and now the time has run out. He is an absconder, a fugitive from justice," Mr Karzai said.
Mr Karzai, speaking from a desert camp near Kandahar, said the Taleban had fled in disarray.
"All of last night they were fleeing the city. Just running away," he said.
Taleban forces are also surrendering in other southern Afghan towns such as Spin Boldak and Lashkargah.
Haji Bashar, a member of Kandahar's new council, composed of tribal elders, clerics and opposition commanders, was quoted by AIP as saying the Taleban had completed their surrender in the city.
He said they had laid down their guns and handed over power to the council.
Other reports, however, spoke of chaos and widespread looting in Kandahar as Taleban fighters abandoned their spiritual capital.
The reports, which have not been independently verified, said no one appeared in control of the city.
Gunfire was said to be heard in many areas and bands of criminals roaming the streets - as were anti-Taleban forces and groups of foreign Taleban fighters, mainly Arabs.
Mr Karzai said the safety of rank-and-file Taleban fighters who had not "associated themselves with terrorism" would be guaranteed.
He denied that the United States had put him under pressure to refuse Mullah Omar amnesty.
Confusion in Kandahar
Mr Karzai said power in Kandahar had been transferred to the new council.
"The real authority... in Kandahar from now onwards should be with the council of tribal elders and religious clergy," he said.
"The objective is to take Afghanistan from rule by the barrel of the gun to rule by the people," he added.
But a prominent anti-Taleban commander, Gul Agha Sherzai, strongly criticised the agreement reached with the Taleban and said Kandahar had descended into anarchy.
"Karzai... has made a very, very wrong decision in Kandahar by himself. He did not consult the elders or anyone else," his spokesman, Khalid Pashtoon told Reuters.
"Now the city is in chaos, there is street-by-street fighting, looting is going on. Everything has been caused by this gentleman, the new prime minister," Mr Pashtoon added.
Bin Laden next target
Thousands of residents have fled Kandahar after weeks of US bombing and a siege by anti-Taleban forces.
Since the US-led military campaign began on 7 October, the Taleban have repeatedly claimed many civilian deaths.
The French news agency AFP quoted an anonymous Taleban source in Pakistan on Friday as saying that US air raids had killed 10,000 people in Kandahar. There has been no independent verification.
BBC correspondents say that with the Taleban's collapse all but complete, the US campaign is now focussing on the hunt for Osama Bin Laden, accused by Washington of masterminding the 11 September suicide attacks in New York and Washington.
There has been speculation that he fled to a mountain fortress - a complex of caves in a part of eastern Afghanistan known as Tora Bora, occupied by hundreds of al-Qaeda fighters.
Units of the Northern Alliance, helped by special forces of the US-led coalition, have attacked the caves and say they now control large parts of them.
They say they have detained many people but have seen no sign of Bin Laden so far.
The US-led coalition's spokesman, Kenton Keith, told the BBC that he believed Bin Laden was still in Afghanistan.