It is understood the family of 19-year-old Duncan Sim, who was last seen in the area of St Andrews Links, has been informed of the discovery.
Mr Sim, 19, has not been in contact with friends or family since leaving an event at Madras Rugby Club in the Fife town at about 11:15pm on Wednesday, 14 March.
The first year chemistry undergraduate was due to arrive at another venue with a group of fellow students.
The alarm was raised the next morning after the teenager failed to return to his halls of residence. Friends described his disappearance at the time as completely out of character.
The remains, found on Sunday, have not been identified and police said inquiries were at an early stage.
A Police Scotland spokesperson said: “At 6.25pm on Sunday June 3 officers from Fife attended West Sands, St Andrews, after the discovery of what is believed to be human remains were found within the water.
“Inquiries remain ongoing at this time.”