Ismail, who lives in the Arghandab district, north of Kandahar, said: "The fighting is still going on with the Taliban. We’ve all left our places ... but maybe there are a few people left behind."
Qari Mohammad Yousuf, a Taliban spokesman, said fighters had captured seven checkpoints around Arghandab.
Al Jazeera's Hamish MacDonald in Kabul says that the Taliban claims to control three-quarters of Arghandab, a district close to Kandahar.
|Al Jazeera aired exclusive video of the|
ongoing battle in Kandahar province
He said: "Control of this area would put the Taliban within striking distance of Kandahar, southern Afghanistan's largest city.
"The city, however, is a major base for the US military and at this stage it is clear the Taliban would struggle to drive out the international forces."
It is the first time since 2001 that Taliban fighters have been able to control the area.
On Tuesday, US-led multinational troops and Afghan forces killed several Taliban fighters, the US military said.
The fighters were killed in Ghazni province, southwest of the capital, Kabul, after they came under fire while conducting a search operation.