Witnesses have told the BBC there has been a build up of Nato-led and Afghan forces in Arghandab district.
"The Taleban are on the move, they can't stay in one place for too long," Afghan Defence Ministry spokesman Gen Zahir Azimi told the BBC.
"They have also blown up a few bridges. We are looking for them everywhere."
Nato helicopters have been dropping leaflets on villages telling people to stay in their houses before an operation.
"Most of the roads and shops are closed, there are check posts everywhere, and military convoys belonging to the Afghan and foreign forces are patrolling the city," Kandahar resident Zahir Jaan told the BBC.
The head of the provincial council, Haji Lalai, said "2,000 families, numbering about 10,000 people" had already left Arghandab.
"They need urgent help. We have talked to UN agencies and other NGOs," he said.
Villagers in Arghandab district contacted by the BBC Pashto service said the Taleban had blown up three bridges and planted bombs under others.
Reports on news agencies say the militants have also been planting mines.
Mullah Daoud, who claimed to be a Taleban commander, rang the BBC from an unspecified location.
He said: "We have gathered in Arghandab because we want to capture an important city like Kandahar."
Mark Laity, Nato spokesman in Kabul, told the BBC there were "indications of Taleban activity which we can't afford to ignore".
But he said the exact level of Taleban presence in the area was "rather harder to determine" at the moment.
Arghandab district lies about 5km (three miles) north-west of Kandahar city and is an important agricultural area.
The Afghan Ambassador to the UN on relations with Pakistan
Taleban leader Mullah Omar used to have a house in the district, which Soviet troops never managed to subdue during years of fighting in the 1980s.
Kandahar is one of the key battlegrounds of the current rebel insurgency against Afghanistan's government and troops from Nato and a US-led coalition.
President Hamid Karzai is from the city and it is also the birthplace of the Taleban.
Friday's mass jailbreak prompted some of the angriest exchanges between Kabul and Islamabad in recent years.
On Monday, Afghanistan's ambassador to Pakistan was summoned to receive a formal protest over remarks by President Karzai.
Mr Karzai had said on Sunday that Afghanistan had the right to send troops across the border to chase militants taking shelter in Pakistan.
The US says cross-border raids from Pakistan are a growing problem.
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