Armed group releases video believed to show American missing since last month.
“Salomi is in good health and he was not hurt during the captivity period.”
The US defence department announced in an earlier statement that Salomi had been returned to US military control on March 25.
A US military spokesman in Iraq declined to comment to AFP about the kidnappers’ claims of a deal involving a prisoner swap.
The Asaib Ahl al-Haq group had issued a videoon February 6 purportedly showing Salomi demanding the release of Iraqi detainees who had “resisted occupation”.
In the video, Salomi called for the prosecution of employees of Blackwater, the US security contractor since renamed as Xe Services, for the killing of unarmed Iraqi civilians in 2007.
He also called for a full and immediate withdrawal of US troops from Iraq.
Salomi’s family were told by Pentagon officials on Thursday that he was safe, but were asked to make no public announcement until Saturday.
His wife, Mura, said she was expecting to be reunited with her husband within a week.
“I didn’t think he’d be alive,” she said. “I can’t wait to hug him and put him in my arms.”
Salomi, who fled Iraq days before US forces invaded for the first Gulf War in 1991, returned in 2007 to work as an interpreter.
The interpreter’s kidnapping was the first of a foreigner in Iraq since Peter Moore, a British computer expert, and his four bodyguards were seized by Asaib Ahl al-Haq in May 2007.
Moore was released last yearbut three of the bodyguards have been confirmed dead, and the fourth is still missing.