The incident happened at about 16:30 BST on Tuesday at a junction in the Moseley area of the city.
It was initially reported to police as a serious road collision but enquiries led the West Midlands force to believe the victim, 35, was deliberately targeted and "left for dead".
A 43-year-old man was arrested on suspicion of attempted murder.
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Police say the Washwood Heath resident has been released "under investigation" while enquiries continue.
The injured man has undergone surgery for a serious head injury and his condition is described by police as critical.
A man was left critically injured after being intentionally driven at in a Birmingham street, police say.