Manvir Singh, 21, from Bishopbriggs, East Dunbartonshire, was last seen in Renfield Street in Glasgow city centre, at about 4.45pm on Wednesday March 19.
The Glasgow University student travelled to St Andrews on the day he disappeared.
A man's body was found outside at Kinkell Braes in the Fife town at around 11am on Thursday morning.
It is believed to be that of Mr Singh, though formal identification has yet to take place.
Police said that a post-mortem will be carried out but that there do not appear to be any suspicious circumstances.
A full report will be sent to the Procurator Fiscal.