The man, believed to be Michael Slaven, was recovered from the river this morning.
Police were called at 9am after reports of a body being found in the water near to the South Portland Street suspension bridge.
DNA tests are taking place to formally identify the body but the family believe it is the missing 23-year-old.
Mr Slaven, from Dumbarton, was reported missing when he did not return home from a night out at 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 at St Andrew's Cathedral in Clyde Street soon after.
His dad John, 52, and mum Elaine, 53, said it was completely out of character for their son to disappear.
Mr Slaven's uncle, Vincent, said the family were contacted by police today with the news that a body had been found. He said their worst fears had now come true.
"We believe the body to be Michael," he said. "He's wearing the same clothes that Michael was wearing when he went missing.
"They're doing DNA tests because of the length of time the body's been in the river. We always feared that the river was the most logical place he might be.
"We retraced his steps and found that he could only have gone a few ways. We were hoping he had gone into the Gorbals.
"Police are now looking at more CCTV pictures to try and piece together what happened."
A spokesman from Police Scotland said: "We can confirm a body of a man was recovered from the River Clyde yesterday morning around 9am near the Suspension Bridge."