The former Manchester United manager is allegedly being considered as a replacement for Niko Kovac.
And Footmercato claims he is one of the frontrunners for the post should the Bavarians axe their current head coach.
Mourinho has been without a club since he was sacked by United in December.
The 56-year-old had been linked with a return to his old club Real Madrid.
However, after seeing Zinedine Zidane beat him to that job, the Portuguese is still searching for his next role.
Now he appears to be in the running for the biggest job in the Bundesliga.
Footmercato state: “According to our information, the Bavarian club would think seriously to move for Jose Mourinho for the next season. The Bundesliga leaders were already interested in the Special One a few years ago, but the opportunity was not created. “
Speaking to The Telegraph last month, Mourinho revealed he has already rejected several offers since his Old Trafford departure.
But he insisted he is happy to bide his time until he finds a special club.
WHAT JOSE WANTS
Mourinho said: “I want to work in a club that understands there is a structure in place.
“I don’t want an internal conflict. I want internal empathy. During my career I have been working in every possible circumstance. The most successful situations are not because of the structure but because of the empathy in the structure.”