The man, feared to be Michael Slaven, was recovered from the river yesterday.
Police were called at around 9am after the discovery near 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.
Michael's uncle Vincent Slaven said: "We believe the body to be Michael.
"He's wearing the same clothes that Michael was wearing when he went missing.
"They are 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 to piece together what happened."
The body was recovered after a high- profile campaign was launched to find Michael, who was reported missing when he did not return home from a night out.
As reported in the Evening Times, Michael, from Dumbarton, 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 images from St Andrew's Cathedral, Clyde Street, soon after.
Michael's dad John, 52, and mum Elaine, 53, along with his sisters Paula, 25, and Monica, 21, and brother Shaun, 20, said it was completely out of character for Michael to disappear.
Officers cordoned off an area of the Clyde Walkway out-side the Gala Casino to the suspension bridge after the body was found.
The pedestrian bridge was taped off, with police standing guard at the cordons.
A search boat with uniformed and plain clothed officers aboard was docked near to the bridge as inquiries were carried out.
MICHAEL'S family had feared the worst when they found his phone was off and his bank card had not been used since he went missing.
On Wednesday family and friends were joined by members of the public to search the St Enoch Square area.
The campaign continued into the night as flyers with the electrician's photograph on it were handed out in a bid to jog people's memories.
Vincent Slaven spoke on the behalf of the family to thank those who helped them in trying to trace Michael.
He said: "The family would like to thank everyone for supporting us in trying to find Michael.
"We would like to thank the Herald and Times Group for helping us so early on and we thank the 300 volunteers for giving up their time and joining us in the search.
"It was heart warming for us to see the love towards Michael. John would also like to thank the local printers who printed the posters for no charge.
"We really have seen such human kindness."
A spokesman from Police Scotland said: "Police Scotland can confirm a body of a man was recovered from the River Clyde yesterday around 9am near the suspension bridge."