The former Manchester City ace is drawing internal criticism from Bayern directors as he struggles to live up to his past reputation.
Blues boss Thomas Tuchel is a fan of the Germany international and could use fringe players to tempt Bayern to part with the midfielder.
Arsenal and Tottenham have also been linked while Julian Nagelsmann is puzzled over how to get the best out of the 25-year-old.
The Blues have the upper-hand, though, and are able to offer Champions League football while being genuine Premier League title contenders.
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