A South Korean from a group of 22 held by the Taleban in south
The office of the governor of Ghazni province and local police also told the BBC the killing had taken place, but neither had seen the victim's body.
The group were kidnapped 10 days ago on the road between
The leader of the group was shot by the Taleban on Wednesday. The militants want government prisoners released.
"We set several deadlines and the Afghan government did not pay attention to our deadlines," the Taleban spokesman said.
The Afghan government is doing all it can to secure the release of the South Koreans, President Hamid Karzai said on Sunday.
He assured a South Korean envoy everything possible was being done to free Christian aid workers - mainly women - who were seized 10 days ago.
Mr Karzai said kidnapping foreign guests was "shameful", un-Islamic and against Afghan culture.
PHOTO CAPTION
Taleban fighters (archive)