He was given trauma care for his head injury by officers at the scene and was then taken to Stepping Hill NHS Hospital, in Stockport, where he remains in a critical condition, the force said.
In a statement on Sunday, GMP said: “At around 9.30pm on Saturday 26 December 2020, officers were responding to a domestic disturbance incident when they were involved in a collision with a pedestrian on Garners Lane in Stockport.
"Police stopped and administered trauma care to a 15-year-old boy until the ambulance arrived. The boy has since been taken to hospital with a serious head injury where he remains in a critical condition.”
The Independent Office for Police Conduct have confirmed they are looking into the circumstances.
A spokesperson said: “After being notified by Greater Manchester Police, we sent investigators to the scene and the police post-incident procedure to begin our enquiries and to start gathering evidence.
“Our thoughts are with the family and anyone else affected by this incident.”
Superintendent Marcus Noden, GMP's divisional commander for Stockport, said: “This is a terribly sad incident and our thoughts are with the young boy and his loved ones, who are being offered support by our specially-trained officers during this distressing time.
“This is an incident that police officers wish to never occur, and support is also being offered to the officers involved in last night's events.
“It is important that a thorough and independent investigation now takes place, and the IOPC will be leading with their enquiries into this incident; therefore it would be inappropriate for us to comment further at this time.”
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