The child and his six-year-old brother were hit by the car on MacDonald Road, Stoke, at about 2pm, West Midlands Police said.
Both were taken to hospital but the younger brother died there.
The six-year-old remains in a critical condition.
A 53-year-old man and a 41-year-old woman were arrested on suspicion of causing death by dangerous driving after a black Ford Focus was found abandoned a short time after the crash.
The road is closed as police speak to witnesses and emergency services attend.
Detective Sergeant Paul Hughes, from West Midland Police’s collision investigation unit, said: “We are still trying to establish how this tragic incident has happened and anyone who hasn’t yet spoken to an officer should call us on 101.
“This is a parent’s worst nightmare and it’s impossible to imagine their grief, but specialist officers will be supporting them as our investigation continues.”
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Information can also be given to Crimestoppers anonymously on 0800 555 111.