The driver was approached by officers at about 03:00 BST on Saturday morning on Brazenose Lane, Stamford.
Shortly after, his Nissan Micra crashed into a wall, killing the driver, 62, who was pronounced dead at the scene.
Lincolnshire Police said the matter has been referred to the Independent Police Complaints Commission.
Brazenose Lane remains closed in both directions due to an accident between East Street and St Paul's Street, police said.
A man has died after crashing into a wall a short time after he was spoken to by police due to the "manner of his driving".