Bamiyan suffered heavily under the Taleban
The graves are situated near the airport of the provincial capital of Bamiyan, about 125 kilometres (80 miles) north-west of Kabul.
One report said the graves were thought to contain at least 35 bodies.
Correspondents say the predominantly Shiite Hazara minority suffered particularly badly during the five years of Taleban rule.
The Taleban - who are Sunni Muslims - were accused of massacring Hazaras in the Bamiyan area in reprisal for earlier killings by Shiite forces fighting against them.
Team heads for Bamiyan
Spokesman Manoel de Almeida e Silva said said the discovery of the graves had been brought to the attention of the UN on Saturday by local representatives.
"We do not yet know the number of bodies or when they were buried. These graves were unknown until now," he said.
The area has been cordoned off, awaiting the arrival of officials from the UN and the Afghan interim government to verify information provided by the local authorities.
The Taleban lost control of Bamiyan in the days before Kabul fell, following numerous battles between opposing forces over the course of a year.
A BBC correspondent who went to the town of Bamiyan in November found it had been totally destroyed by the Taleban before they fled.
Bazaars had been torched in town after town and there were reports of Bosnian-style ethnic cleansing involving the execution of hundreds of local men.
The graves in the Bamiyan area are near the site of ancient Buddhist statues destroyed by Taleban tanks and explosives last year, provoking international outrage.