Manvir Singh, 21, from Bishopbriggs, 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 at Kinkell Braes in the Fife town at around 11am on Thursday morning.
The body 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.
Police and Manvir's family had issued an appeal for information after he went missing.
His father, Harpal Singh, 50, with mum Kirandeep, 41, spoke at a press conference, saying: "Manvir is a much-loved son. All we want to know is that he is alright."
A possible sighting of Manvir on CCTV in Manchester turned out to be false and police appealed to the public to contact them with any information about Manvir's whereabouts.
Manvir was in third year at Glasgow University. His family feared his disappearance could be connected to exam results which he received before he went missing.