BILD says Rudiger, 28, has a year left on his Stamford Bridge deal - he earns around £100,000-a-week and is not even in the club's top 10 earners.
They could lose the Germany international for nothing next summer if he does not sign an extension.
He can also sign a pre-contract agreement with an overseas club in January.
And Bayern have reached out to Rudiger's agent, Sahr Senesie - who is also his half brother.
Rudiger is said to be after £200k-a-week with Chelsea reportedly only willing to pay £150k-a-week.
And that could also be too much for Bayern.
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