One explosion was heard near governor Tooryalai Wesa's heavily guarded compound in the centre of the city and other blasts were heard in other areas in what appeared to be co-ordinated attacks, a Reuters witness and official said.
Shooting erupted after the first explosion and the Reuters witness said he could see gunmen firing from the five-storey shopping mall towards the governor's compound, from where security forces returned fire as helicopters circled overhead.
"The Taliban are attacking the governor's compound and the fighting is still ongoing," Wesa's spokesman, Zalmai Ayoubi, told Reuters from inside the compound.
It was not immediately clear what caused the explosions, although the gunmen were armed with rocket-propelled grenades as well at automatic rifles.
The Reuters witness said he could see black smoke rising near Wesa's compound, which also surrounds the governor's house. Ayoubi said the governor was inside and unhurt.
Three policemen and eight civilians had been wounded and taken to hospital, a provincial health official in Kandahar said.
The Taliban last week announced the start of their "spring offensive", vowing to scale up their attacks against foreign troops and Afghan government officials. Those threats were reissued after the killing of allied al Qaeda leader Osama bin Laden in neighbouring Pakistan.
A Taliban spokesman said the militant group had carried out the attack but said it was unrelated to bin Laden's death.
"A number of fighters are in several locations around the city. These are not retaliatory attacks for the death of Osama bin Laden but are part of our spring offensive," Qari Yousuf Ahmadi told Reuters from an undisclosed location.
Kandahar, the birthplace of the Taliban, has been the focus of military operations over the past year and military commanders have said they have made some security gains, but that these were fragile and reversible.
Violence in Afghanistan reached its worst levels in 2010 since the Taliban were overthrown in late 2001, with record casualties on all sides of the conflict.
Last month, hundreds of prisoners, mostly insurgents, escaped from a jail in Kandahar through a tunnel dug by Taliban militants, which Afghan President Hamid Karzai's spokesman described as a "disaster" for the government.
Days before that, a suicide bomber wearing a police uniform killed Gen. Khan Mohammed Mujahid, the Kandahar police chief.
The Taliban have managed to carry out a number of high-profile attacks inside Kandahar city and in the capital, Kabul, over the past year despite Afghan and foreign forces beefing up security around both cities.
The Taliban issued a statement late on Friday expressing their condolences for the death of bin Laden, who was killed in a U.S. raid inside Pakistan on Monday, but said his death would only revive their fight in Afghanistan.