The driver and passenger of the Jaguar failed to stop following the crash in Wigan at about 16:30 BST on Tuesday, Greater Manchester Police (GMP) said.
According to the police, the boy was thought to have been walking along Old Road in Ashton-in-Makerfield when the car hit him
The 13-year-old was rushed to hospital with serious injuries, where he remains in a life-threatening condition.
A 15-year-old boy has been since arrested on suspicion of causing serious injury by dangerous driving. He remains in custody for questioning.