The Bundesliga club are in the market for a wideman after Ibrahima Traore opted to join Borussia Monchengladbach on a four-year deal when his contract expired at the end of last season.
Stuttgart sporting director Jochen Schneider has revealed that 21-year-old Kostic is on their radar.
Schneider told Bild: "Kostic is one of the most interesting players we have on our list."
Kostic made 38 appearances for Groningen in the Eredivisie last season and chipped in with 11 goals for Erwin van de Looi's men.
He has reportedly attracted the attention of several other clubs across Europe.