The child died after being injured at the David Lloyd health and leisure club.
Police were called to "a concern for welfare involving a child" at the club in Capability Green, in Luton, shortly before 8pm on Thursday.
The girl was taken to hospital, but she died shortly afterwards, police said.
Bedfordshire Police is investigating the incident together with Luton Council's Health and Safety team and the Health and Safety Executive.
In a statement David Lloyd Leisure said: “It is with deep regret that a young child was injured in an incident outside the Clubroom of our Luton club on Thursday and later died in hospital.
“Our thoughts and deepest sympathies are with the family at this time.
"The club is closed, and we are working closely with the relevant authorities while they investigate.”