Rocco Wright was found by his father Steven face-down in the pool of the David Lloyd Club in Moortown, where the family had gone for a swimming lesson in the indoor pool on Saturday morning.
After the three-year-old's death was confirmed, the fitness centre released a statement claiming that members of staff pulled the boy from the pool rather than Steven.
Steven and his wife Catharine say this version of events is 'simply incorrect' and has caused the family further distress.
They have also expressed their shock at comments on social media blaming them for the tragedy and accusing them of leaving Rocco alone while they used the gym or sauna.
Rocco and his four-year-old sister were due to have a lesson, but the family say they were told by staff that not enough instructors were present for it to go ahead. Steven decided to remain in the pool and swim with his children himself, but while talking to the instructor realised his son was missing.
In a statement released through the family's solicitors, he said:
“It was at this point I decided I would forget the lesson and just swim with both my children only to realise I couldn’t see Rocco.
"I checked round the poolside thinking he would be hiding but devastatingly he wasn't. I found him at the far end of pool face down at the bottom. I dived in and got him out onto the poolside and screamed for help. It was only at this point the lifeguard came over and tried to help.
"It was upsetting for David Lloyd to claim a member of staff had 'rescued' him from the pool when that was not even close to what had happened. It was myself that found him and retrieved him from the water.’’
“By making this statement we're not trying to place blame, we just want the facts to be reported accurately.
"We are all totally devastated by the loss of the most amazing happy joyful boy you could ever wish to meet. Rocco truly loved life and we will be left with a void forever that can never be filled. RIP buddy, we'll love you forever.’’
A further statement from David Lloyd Clubs read:
"All of our team at the club are deeply shocked and our thoughts are very much with the boy’s family at this sad time. We are working closely with the relevant local authorities and have launched a full investigation into this tragic incident."