An 18 and a 23-year-old were arrested after the crash in the Ashton Road area of Glass Houghton in the early hours of Sunday morning when they hit another vehicle.
The pair had failed to stop after being flagged down by officers after a violent incident in Selby, North Yorkshire, in which a man was stabbed outside a bar.
They are from the West Midlands area and are still in custody.
A third man, a 26-year-old, who is from North Yorkshire as been arrested on suspicion of attempted murder and possession of a Class A drug. He has been released under investigation.
The stab victim, aged 21, and another man aged 20 were taken to Leeds General Infirmary, the latter having already been released.
The other man remains in hospital and is receiving treatment for his injuries which are not thought to be life threatening.
Detective Inspector Jackie Smart, of North Yorkshire Police’s Serious Crime Team in York said: “We appeal to any taxi drivers or motorists who were in the area at the time who may have dash-cam footage, or members or the public with mobile phone footage, to share this vital information with us.”
Anyone with any information about the incident should contact North Yorkshire Police on 101 and pass information to the force control room quoting the reference 12180144039. You can also email [email protected]