David O'Halloran of Kilmaurs, East Ayrshire was last seen on Friday, January 18, on CCTV at 2.19am as he crossed Henderson Street in Bridge of Allan.
Images show the Skoda, which has been confirmed as a taxi, driving through Bridge of Allan from the direction of Stirling University and then returning a few minutes later.
Superintendent Gordon Dawson said: "We are renewing our appeal for the person who was driving this car or who may have been the passenger to come forward.
"We believe that the car may have dropped off a fare in the Inverallon Road or Minewoods areas."
The car is caught on CCTV at 2.14am as it heads through of Bridge of Allan in the direction of Keir Roundabout.
It is then seen again 2.17am as it returns in the opposite direction back towards the university.
Mr Dawson added: "There is nothing to suggest anything sinister surrounding the car or driver. We are keen to trace the driver, and any passengers, as they may be the last people to have seen David.
"We do not believe that the taxi was from Stirling and we are keen to hear from anybody who may have been driving a white Skoda Octavia 2004-2009 model with a taxi or private hire plate on the right bumper, around this time on Friday, January 18 or even if somebody was dropped off in Bridge of Allan."
The taxi had a small taxi sign on the roof and black strips along the side of the car.
Anyone with information can contact Central Scotland Police on non-emergency number 101 or 01786 456000. Calls can also be made anonymously to Crimestoppers on 0800 555111.