David Lloyd leisure centre on Tongue Lane has been closed after the incident at about 9.45am.
Police were called to the club after a report that a child had entered the pool and drowned.
The boy was taken to Leeds General Infirmary where he was later pronounced dead.
James John-Lewis, a regular at the club, came down from Cheshire to play tennis with a friend and was surprised by how quiet it was.
He said: "It's such a young life, how can something like that happen?
"I'm shocked, I'm gobsmacked. It's a very well-run club.
"It's always busy in the morning.
"Such a young life. My condolences go out to the family. It's very sad news."
Police are treating the incident as a 'tragic accident'.
Mahmoud Moiassari, 58, a member of 15 years, came to pick up his son Ash, 13, who had intended to use the centre.
He said: "It's absolutely terrible. My son's already in shock. So unfortunate, so sad. My thoughts are obviously with the family.
" It's heartbreaking to be honest. My heart goes to the family, losing their kid. You can't comprehend."
His son Ash said: "I just saw lots of police in the building. One of the workers told us a little boy has died in the indoor swimming pool. It's a shock, this has never happened to me before."
Tributes have been left by Yorkshire Evening Post readers on Facebook.
Jane Michelle Cunningham said: "My heart goes out to this little boys family, I'm so very sorry for such a huge loss."
Pam Dowson said: "So sad my heartbreaking. Poor little boy RIP little one God bless you xxxx"
"Struggling to understand how on earth this has happened?!?!?! Poor little lad ," said Richard Bapty.
Peter Hughes said: "RIP little man. Such tragic news"
Michael Packer: "Seriously a sad day for everyone concerned, RIP young man "
travellers. Promoted by Booking.com Detective Inspector James Entwistle, of Leeds District CID, said: “This appears to be a tragic accident and are thoughts are with the boy’s family at this extremely difficult time.
“While there is nothing to suggest that the death was suspicious, we do have a duty to investigate the circumstances of what has happened.
“We understand that there were a number of people in the pool at the time who have left the area prior to police arriving at the scene.
"We would ask these people to come forward to give their accounts of what they have seen and heard.“Anyone who can assist our enquiries is asked to please call Leeds District CID via 101, quoting log 607 of 21/04.”
A spokesman for David Lloyd leisure centre said: "We are deeply saddened by today’s incident at our Leeds club.
"The child was swimming with his family in our indoor pool at the time of the incident.
"He was found unconscious just after 9.30am this morning, and was rescued from the water. Our club teams are fully trained to deal with incidents of this nature, and they immediately administered CPR and the emergency services were called.
"He was taken to hospital by ambulance, accompanied by his mother.
"All of our team at the club are deeply shocked and our thoughts are very much with the boy’s family at this very sad time."