Vlasic has been outstanding for Toro this season, with the Croatia international's form bringing him to the attention of major clubs across Europe.
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And Tuttosport says among the clubs keen are Dortmund.
BVB management see Vlasic as a potential successor for Julian Brandt, who will leave this summer as his contract runs down this season.
Key to the deal could be BVB's Croatian coach Niko Kovac, who is a big admirer of Vlasic.
Vlasic has a deal with Toro to 2027, who made the move from West Ham three years ago.