The German giants are desperate to add some experience to their youthful side and reckon the Croatian schemer fits the bill perfectly.
According to Don Balon, Bayern would be willing to spend £27m to land Modric despite him being 32.
That sizeable fee could appeal to Madrid, where the midfielder remains under contract until 2020.
It has been suggested that Modric would be open to a fresh challenge and also likes the idea of working with former Real boss Carlo Ancelotti again too.
The experienced midfielder has established himself as one of the best players in Europe during his five years in Madrid.