The black Ford Focus hit the tree at the side of Woodside Road, Wyke, with such force that it was left in two pieces, with the engine block ending up some distance further down the road.
Two men, aged 27 and 31, died in the collision which happened at 11.41pm yesterday, West Yorkshire Police said.
It emerged that the vehicle had been spotted by a police car in Manchester Road in the city shortly before the crash.
It is understood it failed to stop for the patrol car when it put on its blue lights.
The Focus drove on and crashed some time later.
The incident has now been referred to the Independent Police Complaints Commission as a matter of routine.
A police spokesman said: “The families of the men have been informed.
“Road closures are in place on Woodside Road junction with Huddersfield Road whilst the investigation continues.
“Any witnesses or anyone with any information is asked to contact the Major Collision Enquiry Team on the non-emergency number 101.”