The 23-year-old is valued at around £12 million and scored five Bundesliga goals for the German outfit last season.
The Mail have claimed that Southampton - who have no manager at the moment following Ronald Koeman's resignation to join Everton - are now speaking with Stuttgart.
However, the report goes on to suggest that Liverpool have also "checked on" the Serbia international as competition heats up for his signature.
With the future of Sadio Mane very much up in the air, could the Saints be targetting Kostic as a potential replacement for the jet-heeled Senegalese star?
The English publication have suggested that Southampton are concerned about the future of Mane, who incidentally has been linked with Liverpool (The Star).
But it remains to be seen who on the South Coast wants Kostic. With Koeman no longer in charge, it begs the question as to who is involved in the reported transfer.
Kostic, who joined Stuttgart in 2014, is contracted to the club until 2019.