The man was driving a Ford Fiesta when it was struck by a Jaguar XF on Friday night.
The driver of the Jaguar XF allegedly failed to stop for officers prior to the crash in Aldridge Road, West Midlands Police said.
Both men were taken to hospital following the collision, which took place just after 11pm.
A 51-year-old man, who was driving the Jaguar XF, is in hospital with injuries not thought to be life-threatening.
He will be interviewed by police at a later date.
The force has since referred itself to the Independent Office for Police Conduct in connection with the incident.
Members of the public who witnessed the crash are being urged to step forward.
Officers are also keen to hear from anyone who may have dash cam footage of the incident, which left a bus stop destroyed.
A West Midlands Police spokesman said: "A 20-year-old man has been seriously injured following a collision on Aldridge Road, Streetly just after 11pm on Friday.
"The Ford Fiesta he was driving was hit by a Jaguar XF, which failed to stop for police prior to this.
"The driver of the Jaguar XF, a 51-year-old man, was also taken to hospital.
"His condition is not life-threatening and he will be interviewed at a later date.
"The incident has been referred to the IOPC, as is standard procedure.
"We want to thank the members of public who stopped and assisted at the scene.
"Anyone who saw what happened or has dash cam footage, and has not already spoken to us, is asked to please get in touch via live chat on our website or call 101 quoting log 4525 of October 7."