The San Siro outfit are desperate to strengthen their squad ahead of the new Serie A campaign and they have earmarked the 23-year-old Brazilian as a valuable addition.
And the attacking midfielder says he would relish the chance to follow in the footsteps of his "idol" Ronaldinho by joining the Rossoneri.
"I am very grateful to Shakhtar for everything they did for me and I will continue to give my all if I stay, but I think I am ready for a big club," the Brazilian told La Gazzetta dello Sport when questioned about a potential move to Milan.
"Ronaldinho has always been my idol and it would be amazing to get the chance to follow in his footsteps. I would not think twice if the right opportunity came up."
Shakhtar coach Mircea Lucescu, meanwhile, has admitted that the Ukrainians are ready to cash in on the player if a good offer comes in.
"Milan have the financial resources and the prestige to sign Douglas Costa," Lucescu told Sky Sport.
"All our players could make a transfer. We will not block anybody. The important thing is that both clubs agree terms and that the player reaches an agreement as well.
"It's not up to me to put a pricetag on him. But we have to bear in mind that we signed these players at the age of 18 and 19 and they became great here at Shakhtar."
Douglas Costa has a contract with Shakhtar until June 2018 but is believed to be valued in region of €25 million.