Khorasan said the people aboard the helicopter included seven Turkish construction workers, an Afghan and two Russian pilots, the BBC reported.
The helicopter, proceeding to Kabul from a NATO base in Khost province, made an emergency landing in bad weather late Sunday in the Azra district of neighboring Logar province, a company official said.
The helicopter was immediately surrounded by Taliban fighters, CNN reported, quoting local officials.
Earlier reports said eight Turkish engineers and an Afghan pilot had been on the flight.
The helicopter was carrying food and other supplies for the NATO-led International Security Assistance Force, the report said.
The Taliban, in a text message to news outlets, said their members had forced the helicopter down, CNN reported.
The report said engineers and others in the country often travel by helicopter to avoid going on roads in areas where there is much insurgent activity.
Local officials and tribal elders were trying to secure the hostages' release, officials said.