Bayern are planning to replace veteran coach Jupp Heynckes at the end of the season.
Bild says Klopp is one of three coaches being considered for the role. The report suggests Klopp is a big admirer of Bayern and so a move would not be deemed impossible.
Germany coach Joachim Low is also being considered, however as things stand, his contract runs until 2020.
It is also suggested the opportunity to appoint former Borussia Dortmund boss Thomas Tuchel has been put on hold.