Three teenagers were also injured, one of them seriously.
The cause was not immediately known, but local newspaper Extra said witnesses alleged the fire started in the air conditioning system.
The Rio de Janeiro mayor's office said the dormitory destroyed by the blaze on Friday was built on land that was not registered for buildings and that the club had no permission to erect lodgings there.
President Jair Bolsonaro, who is receiving treatment for pneumonia, tweeted a statement from the hospital "in solidarity" with the mourning families. "It is a sad day for Brazilian football".
Several of the club's best-known young talents trained there, including midfielder Lucas Paqueta, who made his debut for AC Milan in January, and Vinicius Junior, who joined Real Madrid a year ago.
He added: "Just to remember the nights and days that I spent [at the centre], is to shiver".
"I still can't believe it. Let's pray for everyone". It took around two hours for the fire to be brought under control.More news: Acting Attorney General Whitaker battles angry lawmakers at hearing
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The blaze broke out at 5:00 am in the Vargem Grande district of Rio in a modern facility where the top-flight professional Flamengo squad also trains, the news website G1 said. The statement said an investigation into the licensing process was under way.
"The majority of them [the team] stayed, my friends stayed [at the facility]", he said.
The centre is located in the west of Rio de Janeiro, and according to the city's fire department ten lives have been lost.
Brazilian star forward and Paris Saint-Germain player Neymar posted the club's crest on Instagram, writing: "My sentiments".
The Ninho do Urubu, or Vulture's Nest gets its name from the club's symbol - a vulture.
Brazil football legend Pelé described the club as "a place where young people pursue their dreams".
The training camp is in an area of western Rio that was hit by a fierce storm on Wednesday night.
"While Rio police sometimes kill people in legitimate self-defense, research from Human Rights Watch and other groups shows that many killings are, in reality, extrajudicial executions", the report said.