The man died at the scene on Cottingley Cliffe Road, Bingley, West Yorkshire, shortly after 19:34 BST on Tuesday.
Police said they tried stopping the car after it was seen "acting suspiciously" but it drove off at speed hitting the cyclist a short distance away.
The driver is in custody and the case has been referred to the Independent Police Complaints Commission.
A cyclist has been killed after he was hit by a car as the driver tried to speed away from police.