Costa, who is known for his ball control and mazy runs up the wing, is reported to be looking for a way out of Bayern after a frustrating season under Carlo Ancelotti.
According to Sky Sport 24, Costa's agent, Cesar Bottega, met with Juventus officials in Turin to discuss the player's personal terms.
Costa, 26, reportedly earns 7 million euros ($7.8m) a season at Bayern, who according to the report would aim to sell him for a price in the region of 50 million euros ($50m).
Costa was also on the radar of Inter Milan, who look set to lose Croatian winger Ivan Perisic amid reports of an imminent move to Chelsea.
However he was also a reported target for Liverpool manager Jurgen Klopp, amid existing struggles to land pacey Roma winger Mohamed Salah.