The Brazilian-born player has been afflicted by physical problems since returning to Brazil and could hang up his boots after growing tired of his recent fitness worries
The former Portugal international has sustained seven injuries since arriving at the club on a free transfer from Chelsea in the summer of 2010, the latest of which will rule him out until September.
Deco recently said he plans to see out his contract with Fluminense, but successive injuries have allegedly forced him to weigh up a premature end to his career.
"Deco is extremely upset, he has been allowed not to report until Tuesday," coach Abel Braga said at a press conference. "The player wants to think about this situation. The fans want answers, they want to see him on the pitch, or retire and leave room for someone else.
"But that is not what he wants. Deco is a good guy and he has an amazing character. He has won several important titles. Imagine how hard all this is for him?
"These five days will be good for him to reflect and think thoroughly about his future. It is not just Deco who is sad. We are, too."
Deco, 33, won two Champions League titles, one with Porto and the other with Barcelona, before joining Chelsea in 2008, where he was far from happy.