"Our mujahidin ... have managed to capture the American soldier John Adam after killing a number of his colleagues," said the Mujahidin Squadrons in the undated statement.
It carried a picture appearing to show a US soldier sitting in front of a black banner with a rifle pointed at his head. The authenticity of the claim, which did not say where the man was seized, could not be verified
It was not clear whether the picture had been doctored.
"We will slaughter him in 72 hours if our male and female prisoners in the occupation jails are not released," it said.
Defence officials at the Pentagon said the US military was investigating but had no indication any of its soldiers were missing in Iraq.
A group using the same name, Mujahidin Squadrons, last month claimed responsibility for the kidnapping of a Brazilian engineer in Iraq.
The website claim came 24 hours after US guards shot and killed four Iraqi inmates at Camp Bucca, south of Iraq.
Prisoners began throwing rocks and fashioning weapons after a routine search of one of the camp's 10 compounds, the US military said. Violence then spread to three other compounds.
Troops shot dead four men in the riot which involved hundreds of detainees, the US military said. Six people were injured, five of them by guards. Three of the wounded were taken to a military hospital where they were in a stable condition.