The driver, a Jordanian based in Iraq for several years, was kidnapped in the Saydiyah district of southern Baghdad, Iraqi police said.
"The Jordanian government has started contacts to determine who is responsible for this abduction," a Jordanian official said on condition of anonymity.
Iraqi police said they were looking for two cars thought to have been used in the abduction.
Iraq has been plagued by a resurgent hostage crisis in recent weeks, with fears growing for four Western peace activists and a missing French engineer, who were all snatched in Baghdad.
Jordan has named a new ambassador to Iraq but Ahmad Lawzi has not yet taken up his post in Baghdad where the embassy is run by a charge d'affaires, Suleiman Arabiyat.
In August 2003, four months after the US-led invasion of Iraq, 14 people were killed and 40 wounded in a car bomb attack on the Jordanian embassy in Baghdad, prompting the authorities to relocate to another site.