Baheer Ahmadi, the Kandahar governor's spokesman, said on Thursday that 12 people had been killed, including nine civilians and three police officers. Another 89 were wounded, Presstv Reported.
Police chief Tadin Khan said a car bomb was followed by two other explosions.
After the blasts, gunmen opened fire from nearby positions and members of the security forces were battling them, he said.
"The fighting is over now. Some vehicles were burnt. Now the Afghan army and US helicopters are hovering in the area," Khan said.
Eyewitnesses said following the first explosion, three back-to-back explosions were heard and the gunfight was still going on.
Hayatullah Hayat, the provincial governor of Kandahar, said Afghan security forces had launched a "clean-up operation" to see if any attackers were remaining in the area.
The Taliban said in a statement their fighters had detonated car bombs and clashes were continuing as some fighters had entered the police offices.
Pictures on social media showed a huge plume of smoke rising over the troubled city.
Kandahar was the former seat of the Taliban when they ruled Afghanistan from 1996 until they were ousted in 2001.
The latest assault comes just one day after Taliban operatives disguised as Afghan soldiers fatally shot an Afghan colonel in Ghazni province.
Also on Wednesday, at least 20 Afghan forces members were killed in a Taliban ambush in Abkamari district in western Badghis province.
The US and its allies invaded Afghanistan under the guise of the war on terror in 2001. While the invasion ended the Taliban's rule in the country, it has failed to eliminate the militant group.
Some 18 years on, Washington is seeking truce with the militants as they have boosted their foothold across the country.
Washington is currently engaged in direct talks with the militants, who are against including the Kabul government in the negotiations. The talks have not stopped Taliban from launching attacks.