The highly-rated 20-year-old is also subject to interest from the Manchester City-linked City Football Group.
Evanilson rejected the CFG’s first offer, which would involve him being loaned to their latest club, Troyes, in the French second division.
In the last few days, Crystal Palace have submitted their own offer which Fluminense are looking at.
The Brazilian club are worried about losing Evanilson, an important player for them, in the European transfer window.
However, Crystal Palace are open to loaning him straight back to Fluminense, which would satisfy them.
Evanilson was Flu’s deputy top scorer in 2019 with nine goals in 24 appearances.
Because of his shared ownership, Fluminense would only be entitled to 30% of any fee received for him.