Local media reported that the fire started in a dormitory where youth players sleep.
There was no word yet on the cause of the fire.
The injured were taken to a nearby hospital, firefighters say.
The fire broke out at Ninho do Urubu, a state-of-the-art training centre that was expanded and opened only two months ago.
As well as being one of Brazil's most successful football clubs, Flamengo also has basketball, rowing, swimming and volleyball teams. The fire department have said that they can not yet discount the possibillty that they will find more bodies.
Aerial images from Globo TV showed smoke emerging from a charred building.
Jefferson Rodrigues, who runs a small inn near the club, said he had reached a 15-year-old player he had befriended.
He added: "Just to remember the nights and days that I spent [at the centre], is to shiver".
Flamengo is one of Brazil's biggest and best-known clubs.
"The kids from the Flamengo youth team were sleeping there at the time", a firefighter told Brazilian news network GloboNews. The development leagues identify promising players at a young age, working with them as they grow through their teenage years. Many players stay at the facilities while training.
"What sad News! Praying for everyone!"