The group, calling itself Mujahideen Squadrons, said in a statement posted on an Islamic website that they are holding John Adam and will kill him within 72 hours if Iraqi prisoners are not freed.
The site also had picture showing the hostage held at gunpoint with his hands tied behind him.
"God willing, we will behead him if our female and male prisoners are not released from US prisons within the maximum period of 72 hours from the time this statement has been released," the statement said.
The group also claimed they also killed several other US soldiers. The authenticity of the undated statement could not be verified.
Meanwhile, Iraq's interim president Ghazi al-Yawer said it would be complete nonsense to ask foreign troops to leave now, though some could depart by the end of the year.
Asked during a news conference whether the presence of foreign troops might be fuelling the Sunni Arab revolt by encouraging rebel attacks, he said: "It's complete nonsense to ask the troops to leave in this chaos and this vacuum of power."
Foreign troops should leave only after Iraq's security forces are built up, the country's security situation has improved and some pockets of terrorists are eliminated, he said.
"By the end of this year, we could see the number of foreign troops decreasing," said al-Yawer, who had earlier criticized some aspects of the US military's performance in Iraq.
