Brazil winger Costa had been expected to leave the Allianz Arena in this window, having made just 14 Bundesliga starts under Carlo Ancelotti last term.
Juventus have reportedly struck a loan deal to bring Costa to Serie A.
The former Shakhtar Donetsk man is to become the latest player to transfer between the German and Italian champions, a list which includes Arturo Vidal, Kingsley Coman and Medhi Benatia, who made a loan deal from Munich permanent in this window.
And the Morocco international looks like playing a key role in helping Costa settle into Serie A, having offered him a warm welcome on Twitter.
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