Governor Raja Nawaf Farhan al-Mahalawi was captured as he drove from al-Qaim to the provincial capital of Ramadi on Tuesday morning, said his brother Hammad.
The kidnappers later telephoned the family and said they were holding the governor until US forces pull out of the Syrian border town of al-Qaim, Hammad Nawaf Farhan al-Mahalawi said.
"The kidnappers have demanded that American forces leave al-Qaim in order to release him," he said.
Lieutenant-Colonel Steven Boylan, a spokesman for US forces in Iraq, said: ''We don't respond to insurgent or terrorist demands.''
Major US offensive
US forces are conducting one of their largest offensives in six months in the remote desert region.
The US command said as many as 100 fighters were killed in the first 48 hours of Operation Matador, as American troops cleared villages along the meandering Euphrates then crossed in rafts and on a pontoon bridge.
"We don't respond to insurgent or terrorist demands"
spokesman for US forces in Iraq
At least three marines were reported killed and 20 wounded during the first three days of the offensive, the biggest US operation since Falluja, six months ago.
Al-Mahalawi only recently became governor after tribal leaders forced out his predecessor Faisal Raikan al-Gut al-Nimrawi, who narrowly escaped a roadside bombing in February.
Al-Mahalawi, who is originally from al-Qaim, served as mayor of the town under Saddam Hussein.
Speaking to Aljazeera from the Iraqi town of al-Qaim, 340km west of Baghdad, director of al-Qaim hospital Dr Hamdi al-Alusi said: "I confirm that most casualties are civilians, namely elderly men, women and children."