Mulla Latif Hakimi offered no proof to back up the claim on Friday but he said a video would be released of the soldier's death. "This American will never be forgiven. Definitely he will be killed."
Hakimi said he was last in contact on Wednesday with the rebels who he said were holding the American and was told the captive's health was good and that he had not been abused.
He said the US soldier was being held in a house in Kunar.
Hakimi often claims responsibility for attacks on behalf of the Taliban, but his exact link to the Taliban leadership is unclear.
The US commando is the last of a four-member US Navy Seal team missing for 10 days in Kunar province, near the Pakistani border. One of the men was rescued and the other two have been found dead.
Search under way
US military spokeswoman Lieutenant Cindy Moore responded by saying she hoped the commando was "not in harm's way" and said the army was making every effort to locate him.
"This American will never be forgiven. Definitely he will be killed"
Mulla Latif Hakimi,
purported Taliban spokesman
About 300 troops and several aircraft are searching in the rugged mountains in eastern Afghanistan, Moore added.
The eastern province has long been a hotbed of rebel activity and a haven for fighters loyal to renegade former premier Gulb al-Din Hekmatyar, who is wanted by the United States.
US officials said al-Qaida fighters also were in the region. Osama bin Laden was not said to be there - although he is believed to be somewhere along the Afghanistan-Pakistan frontier.
The region's wooded mountains are popular with fighters because they are easy to infiltrate from neighbouring Pakistan and offer plenty of places to hide.
On Saturday, a US air strike in the region killed as many as 17 civilians, prompting a strong rebuke from the Afghan government.
Hundreds of US soldiers are
searching for the missing soldier
A special forces helicopter carrying reinforcements to the mountain area crashed on 28 June, killing all 16 Americans on board, the deadliest single attack on the US military since the war began in 2001.
The claim by the Taliban follows an unprecedented spate of insurgent violence that has left about 700 people dead.
Afghan officials insist the violence will not disrupt landmark legislative elections set for September.
Elsewhere in Afghanistan, suspected Taliban rebels late on Wednesday attacked a government office, about 64km south of the Afghan capital, Kabul, local police chief Khan Muhammad said.
Police guarding the building fought back during a two-hour gun battle before the rebels ended their attack. No officers were killed, he said. The fighting was the closest to Kabul by suspected Taliban rebels in months.