Serie A side look set to submit an offer to the German club.
Sttutagrt were relegated from the Bundesliga to the Zweite Bundesliga after finishing 17th in the league table last season, which could cause a number of their players to depart.
One of those names is 23-year-old Kostic who, according to German sports site Kicker, has a value of around €10 million and has been targeted by Inter.
The Italians will have to move quickly, however, as Kostic has also caught the attention of fellow Germans Hamburg and Spanish side Valencia.
The player moved to Stuttgart from Groningen in 2014 and has since managed eight goals in 63 games, assisting a further 14.
Apart from being ruled out of four games towards the start of the season due to injury, Kostic played a part in every match of Stuttgart’s Bundesliga campaign last season.
