The pedestrian is reported to have sustained a head injury but left the scene before emergency services arrived.
Officers are urging a pedestrian involved in a collision in Coalville to seek medical assessment.
Police received a call at 6.47 pm on Wednesday 15 January from East Midlands Ambulance Service following their attendance at a road traffic collision on A511 Stephenson Way.
The collision involved a blue Mini Countryman and a pedestrian.
The pedestrian is reported to have sustained a head injury but left the scene before emergency services arrived.
The driver of the car was not injured.
Sergeant Kyle Taylor said: “We suspect this man may have sustained a serious head injury and we would urge him to seek medical assessment urgently.
“Enquiries have been carried out in the surrounding area, but we have been unable to trace him.
“He is described as wearing a black gilet, black top and blue tracksuit bottoms and spoke with a foreign accent.
“We are clearly concerned for this man’s welfare, and we would strongly urge he attends a hospital or an urgent care centre to be checked over.
"As our enquiries continue into the collision, anyone who may have witnessed the collision or the events leading up to it and hasn’t spoken to us is also asked to get in touch.”
Anyone with information is asked to contact police on 101 quoting incident 511 of 15 January or visit us online at www.leics.police.uk