Police were called to the David Lloyd club in the city’s Tongue Lane shortly before 9.45am on Saturday after reports that a child had drowned.
The boy was taken to Leeds General Infirmary, where he later died.
Officers are treating the death as a tragic accident but want to speak to anyone who was in the pool at the time.
Detective Inspector James Entwistle of West Yorkshire Police in Leeds said: “While there is nothing to suggest the death was suspicious, we do have a duty to investigate the circumstances of what has happened.
“We understand 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.”
The sports centre told the Press Association it was closing for the day following the incident and would be making no further comment.
Anyone with information is asked to contact police.