The former Rangers, Blackpool and Liverpool midfielder, who turned 36 last week, was arrested in the early hours of Monday morning on Dundee Road West.
It is alleged that the 26-time capped Scotland international was driving under the influence of alcohol when he lost control of his vehicle and collided with a tree.
A Police Scotland statement read: "Shortly after 4.25am on Monday 13 December, a 36-year-old man was arrested and charged for a road traffic offence in Dundee Road West, Dundee.
"A report will be sent to the Procurator Fiscal."
Adam will miss Dundee's cinch Premiership match against Hibernian at Easter Road this evening after going off injured in Saturday's defeat away to Ross County.
He joined the Dens Park outfit in the summer of 2020 and helped them achieve promotion to the top flight last season.
A Dundee spokesperson told Sky Sports: "We are aware of the incident but will be making no further comment."