The claim's authenticity could not be immediately verified.
The Islamic Army in Iraq on Thursday said it had killed "the American security consultant for the Housing Ministry," after the United States failed to respond to its demand of the release of Iraqi prisoners.
A video issued by the group was aired on Tuesday on the Aljazeera Arabic satellite channel showing the hostage - identified as Ronald Schulz, 40, an industrial electrician from Alaska - sitting with his hands tied behind his back.
Thursday's statement, posted on an Islamic rebel Web forum, did not name the hostage and provided no evidence he had been killed, but said pictures of the slaying would be released later.
In the rebel tape, aired by Aljazeera two days back, the group had said it was giving the US military in Iraq 48 hours to release all Iraqi prisoners, otherwise it would kill him.
The group also demanded the US army compensate the people of Anbar governorate for the losses they had suffered from the frequent US attacks on their cities. Both Falluja and Ramadi are in Anbar province and have come under heavy US fire.
Last year, a group with the same name killed Italian journalist Enzo Baldoni after Rome refused its demand to pull troops from Iraq.
The kidnapping comes just days after armed men seized four Christian peace activists, two Canadians, a Briton and an American, as well as a German archaeologist and a French engineer in Iraq.
Thousands of civilians have been kidnapped in Iraq since the fall of Saddam Hussein's government, including more than 200 foreigners.
Many hostages have been released, but about 50 have been killed, some by beheadings broadcast on the internet.