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.

Alexander Beltran Herrera's unit was responsible for moving the hostages and imprisoning them in chains.
The contract workers were on an anti-drug surveillance mission when their aircraft broke down in the Colombian jungle in February 2003.
They were eventually rescued by the Colombian military in 2008.
Hostages Thomas Howes, Keith Stansell and Marc Gonsalves were imprisoned by members of the Revolutionary Armed Forces of Colombia (Farc) and held by Herrera's unit for around two years.
While Alexander Beltran Herrera did not have much contact with the hostages, the court heard how the three men were taken deep into the jungle and kept in chains for much of the five years of their captivity.
Keith Stansell removed his shirt to show the court scars around his neck from the chains and described being chained to a tree whilst suffering from malaria.
But, addressing Herrera, he said: "I don't want him to suffer like we did."
Herrera was extradited to the United States in 2012.  His lawyer described him as a man of little education who had been trying to build a new life after he deserted from the Farc in 2009.
According to Associated Press, he apologised to the men during the trial, saying: "I feel shamed about what you all had to go through.''
Colombian military forces rescued the three Americans in July 2008 in a high profile helicopter rescue operation, which also brought about the release of French-Colombian politician Ingrid Betancourt, as well as 10 other hostages.
The Farc are currently engaged in dialogue with the Colombian authorities in a bid to negotiate a peace deal - some see it as the best chance for a negotiated settlement since the guerrilla group launched its armed struggle in 1964.
A man was left critically injured after being intentionally driven at in a Birmingham street, police say.
[NEXT_CONCEPT]
A US judge has sentenced an ex-Colombian rebel to 27 years in prison for his role in the kidnapping of three American contract workers in 2003.