A leaflet distributed by the U.S. military to civilians in Logar province, east of Kabul, on Sunday shows an unidentified missing U.S. navy sailor, with an offer of $20,000 US for information on his whereabouts. (Associated Press)

The Taliban now claims to have captured one U.S. navy serviceman and killed another after an ambush in Afghanistan's Logar province.

Taliban spokesman Zabiullah Mujahid told The Associated Press on Sunday that the American was killed during a brief gunfight in an area of Logar province that is under insurgent control, while a second American was captured.

Adm. Mike Mullen, chairman of the U.S. Joint Chiefs of Staff, declined to comment on the report and would only say that a search was continuing for the missing Americans.

NATO has confirmed the two failed to return to a military compound in Kabul on Friday.

The U.S. military has been broadcasting radio messages offering $20,000 US for information leading to their safe return.

The Taliban has reportedly contacted Afghan officials, offering to exchange the body of the dead soldier for the release of several insurgents.

Afghan police said they saw the Americans driving in an armoured SUV, which failed to stop when police tried to wave it down to warn them of Taliban activity in the area.

The Americans went into a bazaar. When they drove away, they were ambushed, police said.

Their disappearance comes amid warnings from Afghan security officials that Taliban insurgents are initiating a campaign of kidnappings and assassinations in response to the ongoing NATO surge.

Specialist Bowe Bergdahl of Hailey, Idaho, was captured by the Taliban in Paktika province in June 2009. He is the only known American in the Taliban's hands and has appeared in videos posted on Taliban websites.

With files from The Associated Press