The club recently tied him to a new contract, increasing his buyout clause from £5million to £21million, and agent Cesar Bottega says Italian clubs are also monitoring his progress.
"Europeans already knew Douglas from the Brazil (youth) team, but still they needed to see him perform at senior level.
"Against Santos and Botafogo, he proved that already he has the ability to play among professional."
Bottega is keen for any move to come after Gremio have played in the Copa Libertadores - South America's equivalent of the Champions League - which takes place in the first half of next year.
But with Gremio in debt to the tune of about £45million, his club may be forced to cash in.
"Douglas still needs to ripen, and I would not like it if he left now,' said Bottega.
"But when you are negotiating with such markets as the Italian, English and Spanish, it is not always possible to stick to our wishes. The pressure is great."