Juventus, Real Madrid, Barcelona and Manchester United are all interested in the 18-year-old, nicknamed The Black Pearl.
“We know that many people from various European clubs have come to Brazil to see Douglas,” his agent Cesar Bottega told Calciomercato.
“Some have seen him play for the Under 20 national side and since then we have been unable to declare our public appreciation.
“He showed many qualities and it is normal people will like him.
“But we have not talked to any teams, we know clubs are following him but there is nothing official.
“Nobody has done anything, the big clubs are very careful. They know who to watch – but there is nothing.”
Costa has a massive £21m release clause in his contract with Gremio and it is believed the Brazilian side is not willing to sell him for less.
Juventus want to bring in Costa as a replacement for veteran winger Pavel Nedved.