The incident happened around 2pm on Sunday 24th July on Bogwood Road, Mayfield.
A silver or light blue people carrier mounted the kerb near to Easthouses Church and struck a 23-year-old man who was walking on the pavement.
The car then drove off towards Lawfield Road without stopping.
Following the collision the victim sustained minor injuries to his knee and hip but did not require medical attention.
Officers are now asking anyone who can assist with their investigation to come forward.
Sergeant Alan King from Dalkeith Police Station said: “Fortunately, the victim was not seriously hurt as a result of this incident and we are currently pursuing various lines of inquiry to trace the driver of the people-carrier.
“We believe there were a number of people waiting at the bus stop opposite David Scott Avenue at the time of the collision and they may have witnessed what happened. I would ask these individuals to contact police immediately.
“In addition, anyone with any further information relevant to this investigation should also get in touch.”
Those with information can contact Police Scotland on 101 or alternatively, the charity Crimestoppers on 0800 555 111.