There was no confirmation that a military operation had begun in Ghazni province, where the Christian volunteers were abducted nearly two weeks ago by the Taliban.
The Islamist insurgents, who are demanding the release of rebel prisoners, killed two men from the original group of 23 after previous deadlines they had set expired.
“The hostages are alive,” said Khowja Seddiqi, district chief of Qarabagh. “The national army has dropped leaflets from helicopters telling people in several districts to evacuate their houses because it wants to launch an operation.” Afghan army and foreign troops were stationed in the area, but no operation to rescue the hostages had begun, Seddiqi said.
A Taliban spokesman also said the hostages were alive, and that there was no sign of a bid by the military to rescue them.
Earlier, the Defence Ministry said the Afghan National Army had launched an operation in Ghazni, but insisted it was “routine” and was not linked with the kidnapping.
The BBC said fighting had erupted in the area where the hostages were being held.
Earlier in the day, a Taliban spokesman said the group was expecting to hear from Afghan mediators over its demand for the government to free rebels from jail, but insisted some of the hostages would be killed if that demand was not met by 0730 GMT.
The Afghan government has said that giving in to rebel demands would only encourage more kidnapping.