The US military said it had no information to support the claim.
"We killed him at 11 o'clock today; we killed him using a knife and chopped off his head," Taliban spokesman Abdul Latif Hakimi said on Saturday from an undisclosed location. He said that the body had been dumped on a mountain in the eastern province of Kunar.
The US military has said it has no information to suggest the Navy Seal commando, part of a four-man team that went missing during a clash with militants in mountainous Kunar on 28 June, has been captured.
Asked about the Taliban claim that the man had been killed, US military spokeswoman Lieutenant Cindy Moore said: "I don't have any information on that."
Hakimi, whose information has often proved unreliable in the past, said the body of the soldier had been left on the top of a mountain in Kunar's Shegal district.
"He is wearing red clothes," he said. "We got the information we wanted from him during the interrogation."
The US military found two of its
missing commandos dead
Wali Allah Shahin, Aljazeera's correspondent in Afghanistan, said Hakimi had previously threatened the Taliban would kill the US soldier, adding that they had a Fatwa [Islamic decree] by Afghan religious scholars allowing the killing of any prisoner.
"We are in a state of war and will kill any American we hold captive", the Hakimi said.
The Pakistan-based Afghan Islamic Press (AIP) news agency also quoted a Taliban commander in Kunar, Mohammad Ismail, as saying that the commando had been killed.
AIP quoted Hakimi as saying the killing followed a decision by the Taliban's "council of religious leaders."
Killed in action
The US military said two of its missing commandos were found dead on Monday, having been "killed in action", while another had been rescued and one was missing.
A US helicopter sent to help the team was shot down the same day the team went missing during a battle with fighters, with the loss of all 16 troops aboard. These were the US forces' heaviest loss in a single combat operation since they overthrew the Taliban in late 2001.
Hundreds of US soldiers, backed by Afghan troops and helicopters, have been searching for the missing commando in Kunar for the past 12 days.
Agha Jan, a senior pro-government cleric, was killed on Saturday in the southeastern province of Paktika.
Jan was stabbed to death along with his wife at his home on Friday evening, Paktika Governor Gulabuddin Mangal said.
Islamic cleric Agha Jan was killed
in Paktika, Afghanistan
"It was the Taliban who did it," Mangal told Reuters, adding that four suspected Taliban members had been captured in connection with the killings.
Taliban officials could not be immediately contacted for comment.
Jan, the head of the influential Ulema Council in Paktika, had preached in favour of President Hamid Karzai's Western-backed government and US-led forces, who overthrew the fundamentalist Taliban government in late 2001, Mangal said.