The 23-year-old has in lieu of completed a move to Hamburg after Stuttgart were relegated from the Bundesliga for the foremost time in 41 years last season.
German publication Bild asked Liverpool were interested in bringing the Serbia international to Anfield after scouting him for the before two years.
But Kostic, who scored five goals and set up a further eight in all contests last term, has opted to remain in Germany after finalising a five-year grapple with with Hamburg.
“We are glad that we were able to realise the remove, he is a player with a high tempo in his actions,” chairman Dietmar Beiersdorfer advised the club’s official website.
“Despite being only 23 years old, he has already a lot of happening, knows the Bundesliga and has proven his qualities in the st two years. Filip also continues to be a high development potential.”
The winger is understood to have cost Hamburg about £14million, having initially cost Stuttgart just half that rear in 2014.