Dubai-based al-Arabiya network said it received the tape from a group calling itself the Islamic Rage Battalion, which said it had kidnapped the man on 3 May.
He was shown on Thursday blindfolded with a chequered Arab scarf and wearing a jacket.
"My name is Aban Elias from Denver, Colorado," the captive was quoted as saying.
"I am a civil engineer working in Baghdad ... and we are working with the Pentagon ... I was kidnapped and I call upon Muslim organisations to intervene to release me."
The network said a copy of the man's US passport showed he had been born in Iraq. Al-Arabiya said his captors had not made any demands.
Elias's mother in Denver broke down when she heard the news.
"Please free up my son. He has no relation with anybody. He only went to work in Baghdad," Thamara Mahmoud al-Saloum said outside the family's home.
The man's brother Kazwan Elias told reporters his brother had no political affiliations in Iraq and is an American citizen.
"I'm calling upon all the Islamic organisations to free him. He's got three little kids, one is one-year old, a four-year-old and a six-year-old.
Aban Elias's wife and children are living in Jordan.
US State Department spokesman Richard Boucher confirmed authorities had been in contact with his family.
"The US embassy in Amman and (the) US consular officer in Baghdad are working to ascertain the welfare and whereabouts of this US citizen," Boucher told reporters in Washington.
Dozens of Westerners, most of them working for the US-led occupation, have been kidnapped in Iraq recently. Some have been released and others killed by their captors.