At around 4pm on May 20, a black Ford Fiesta failed to stop for officers on Loxley Road, Hillsborough. It was then pursued for nine miles before the driver got out on Mortimer Road in Midhopestones.
During the pursuit, the vehicle collided with a low loader truck on Penistone Road, and narrowly avoided hitting a white Ford Fiesta on Mortimer Road.
Officers are now urging witnesses to the incident to come forward.
They also want to trace the drivers of the truck and the white Fiesta.
Two men, aged 21 and 29, were arrested on suspicion of driving, drugs and weapons offences. They have been released under investigation while enquiries continue.
Anyone with information should call South Yorkshire Police on 101 quoting incident number 687 of May 20.
Anyone with dashcam footage can email it to [email protected] quoting the incident number in the subject line.