The Daily Star says Klopp, 49, is now regarded by the German giants as the ideal successor to current coach Carlo Ancelotti.
That news will come as a massive blow to Anfield chiefs, who handed Klopp a new bumper SIX-YEAR deal earlier this summer.
Not that they need worry about losing their hugely popular and charismatic German leader any time soon.
That situation suits Bayern perfectly. They are very happy with Ancelotti and expect the Italian to stay until the end of 2017-18 season at least.