Officers want to speak to anyone who witnessed a collision just before Cheltenham Half Marathon earlier this month.
The incident happened near the Wellington Road closure that was in place at around 8.30am on Sunday 3 September.
It has been reported that a man driving a black Volkswagen Golf became involved in an argument about driving through the road closure. His vehicle was then in collision with a man before turning around and travelling away from the scene.
The pedestrian was left with muscle damage and had to attend hospital.
The registered owner of the suspected vehicle is being contacted but it is believed there were other witnesses at the scene and officers need to speak to them.
If you were the driver or a witness you can call 101 or information can be provided online by completing the following form and quoting incident 320 of 9 September: www.gloucestershire.police.uk/tua/tell-us-about/cor/tell-us-about-existing-case-report/