La Stampa says Toro have offered €5m for the teenage Scotland international.
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Miller missed the weekend's Premiership clash with Rangers, with injury cited. However, Turin sources say his absence was due to negotiations now being underway.
Miller was named the Premiership's Young Player of the Year last season, with four goals and six assists in 34 appearances.
At Toro, should a deal be struck, he would join Scotland teammate Che Adams, with the striker having made the move a year ago from Southampton.