The US has sent more ground troops into Afghanistan
Anti-Taleban Afghan fighters are reported to be closing in on the city as efforts to negotiate the surrender of militia commanders continue.
Many residents of Kandahar, including Taleban officials, have already fled.
Khalid Pashtoon, a spokesman for former Kandahar mujahideen governor Gul Agha, told the Reuters news agency that Agha's army of 3,000 fighters was massed six kilometres (four miles) south of the airport but had no immediate plans to advance.
"The situation right now is quite quiet and calm. Bombing is going on right now, but there is no fighting," Pashtoon told Reuters by satellite telephone.
A top commander of the anti-Taleban Northern Alliance, speaking in Kabul, earlier claimed that alliance forces had entered Kandahar.
But there has been no independent confirmation of this, and correspondents point out that the Northern Alliance, mainly made up of ethnic minority Uzbeks and Tajiks, would be unwelcome in the primarily-Pashtun south.
Speaking in Washington, Rear Admiral John Stufflebeem, the deputy director of operations for the US Joint Chiefs of Staff, said he could not confirm the report.
Taleban defiance
Taleban leader Mullah Mohammed Omar has told his fighters that there is no question of surrender, and no need for negotiations.
As well as Kandahar, the key border town of Spin Boldak remains in Taleban hands.
"Don't vacate any areas," Mullah Omar said. "This is not a question of tribes - this is a question of Islam."
There have been reports of secret talks between the Taleban and tribes in the area, but these have been publicly denied.
The former governor of Kandahar, Gul Agha, is in control of a village between the city and the border, backed by anti-Taleban forces, and clashes have been reported.
Pressure on al-Qaeda
US officials say anti-Taleban forces have captured a follower of Osama Bin Laden who is also related to Sheikh Omar Abdel-Rahman, the Egyptian cleric jailed in the US for plotting to blow up New York City landmarks.
The Sheikh's son Ahmed was reportedly being held in Mazar-e-Sharif where he was being interrogated by American agents. The US believes he is a recruiter for Bin Laden's al-Qaeda network.
US sources also said there were "credible" but as-yet uncorroborated intelligence reports that a Taleban intelligence chief had defected to the Northern Alliance
At Thursday's Pentagon briefing in Washington, spokeswoman Victoria Clarke refused to comment but said defections among senior Taleban officials were "another sign that the Taleban continues to be degraded".
Troop moves
US Marines are continuing to arrive at the Dolangi airfield base about 130 kilometres (80 miles) south of Kandahar, increasing the force to about 800, said Pentagon spokeswoman Victoria Clarke.
Their role is reported to be the closure of potential escape routes to Pakistan and Iran for Taleban and al-Qaeda fighters, and the Pentagon said that their number was likely to increase.
The BBC's Steve Kingston says they will also prove crucial in searching the region's myriad of caves and tunnels for Osama Bin Laden, in what may prove a far more dangerous aspect of the campaign.
US Defence Secretary Donald Rumsfeld described the work as part of "dangerous" yet necessary tactics.
"Our efforts, of course, will be shifting from cities at some point to hunting down and rooting out terrorists where they hide... this is difficult work, it's dangerous work," he said during a visit to US Central Command in Florida.