The Taliban said last week they were holding the soldier, who the U.S. military earlier described as possibly being in enemy hands.
Abdullah Jalali, a spokesman for Taliban commander Mawlavi Sangin, told The Associated Press in a phone interview the soldier was healthy. He said the soldier would be killed unless the U.S. stops airstrikes in Ghazni province's Giro district and Paktika province's Khoshamand district. Still, Jalali said the final decision about the soldier's fate will be made by Taliban leader Mullah Omar.
Jalali said Giro has been heavily bombed by international troops but did not otherwise explain why they chose those areas.
The U.S. military said the soldier was noticed missing during a routine check of the unit on June 30 and he was "believed captured." The Taliban confirmed on its Web site on July 6 that it was holding the soldier.
"Five days ago, a drunken American soldier who had come out of his garrison named Malakh, was captured by mujahedeen... He is still with mujahedeen," said the report. The short Web message did not elaborate on his whereabouts or their plans for him, nor did it provide any proof.
The U.S. military earlier said it intercepted communications in which insurgents talked about holding an American.
His body armor and weapon were found on the base, and U.S. defense sources said the soldier "just walked off" post with three Afghans after work. They said they had no explanation for why he left the base.
Earlier Thursday, a suicide bomber rammed an explosives-laden station wagon into a police convoy in southern Afghanistan on Thursday, killing three officers, the Interior Ministry said.
Four other police officers were wounded in the morning attack in Nimroz province's Khashord district, the ministry said in a statement.
Militants regularly use roadside and suicide bombings to attack international and government troops in Afghanistan, making the makeshift explosive one of the biggest threats to forces trying to rout the resurgent Taliban.
The bombing follows an attack on an international forces supply convoy in southern Paktika province on Wednesday that left at least eight insurgents and two police officers dead, along with a private security guard, the ministry said.
A provincial official put the death toll from the Paktika clash much higher, saying 21 insurgents and three border police died. Hamidullah Zhwak, a spokesman for the provincial governor, said he had no reports of private security guards killed.
Military supply convoys in Afghanistan are operated by contractors and guarded by private security guards. Zhwak said there were about 80 guards protecting the convoy that was attacked.
Two Afghan army soldiers were killed in two other attacks in the south on Wednesday, the Defense Ministry said.
In the east, meanwhile, international and Afghan forces killed two insurgents who helped mount bomb attacks in the area, NATO forces said. Four other militants were captured in Wednesday's operation in Kunar province, the military alliance said in a statement. It did not give further details.