Michael Slaven's family have been informed of the discovery of the body, which was recovered from the river yesterday morning, and DNA tests are now taking place to help with formal identification.
Police were called to the water near to the South Portland Street suspension bridge at around 9am yesterday. It followed a series of appeals by relatives of Mr Slaven, 23, who left the Sugar Cube nightclub in Glasgow's Queen Street shortly after 2am last Friday.
CCTV images showed him in a Subway sandwich shop before acting erratically at around 2.30am in nearby St Enoch Square.
There were claims he had been seen on CCTV at St Andrew's Cathedral in Clyde Street soon after.
Michael's father John, 52, and mother Elaine, 53, along with his two sisters and brother had feared the worst when they found his phone was off and his bank card had not been used since he went missing.
Relatives and friends were joined by members of the public to search the area around St Enoch Square and hand out flyers on Wednesday.
Officers cordoned off an area of the Clyde Walkway outside the Gala Casino to the suspension bridge after the body was found.
The pedestrian bridge was taped off at each end, with police standing guard at the cordons. A search boat with uniformed and plain clothed officers aboard was docked near to the bridge.
A spokesman from Police Scotland said: "Police Scotland can confirm a body of a man was recovered from the River Clyde around 9am near the Suspension Bridge."