A senior Afghan official says Nato and Afghan forces have cleared out Taleban militants from the outskirts of the southern city of Kandahar, reports BBC.
Kandahar governor Assadullah Khalid told reporters that Taleban fighters had fled Arghandab district.
He said the Taleban had suffered heavy casualties in the fighting. Nato has not confirmed the report.
Hundreds of people fled their homes in Arghandab district ahead of fighting between troops and militants
The BBC's Martin Patience in Kabul says the Afghan government is under pressure to prove its effectiveness in dealing with militants in Kandahar after hundreds of militants escaped from a jail there last week.
Nato spokesman Mark Laity told the Associated Press news agency that the military operation in Kandahar was progressing "methodically and successfully" and meeting very little resistance.
He confirmed some overnight airstrikes by Nato planes in the area, and said he did not have casualty figures.
Governor Assadullah Khalid, however said the Taleban had "suffered hundreds of dead and wounded and many of their casualties are Pakistanis".
A spokesman for the Afghan defence ministry told the BBC Wednesday that more than 20 Taleban fighters had been killed in an air strike close to Kandahar, but there has been no independent confirmation of this.
The Afghan defence ministry had estimated that around 300 to 400 militant fighters were operating in Arghandab district and had taken at least eight villages.
On Friday about 350 Taleban fighters escaped with other inmates from a jail in Kandahar. Only a handful of prisoners have been recaptured.
The BBC's Martin Patience in Kabul says it is not clear whether some of those militants are being targeted by the current offensive.
Our correspondent adds that the jail raid was a major coup for the Taleban, which draws much of its support from the south of the country.
Arghandab district lies about 5km (three miles) north of Kandahar city and is an important agricultural area.
It offers good fighting positions for Taleban militants with its vineyards and pomegranate orchards and numerous irrigation ditches.
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.
