Blues boss Antonio Conte is looking to strengthen his Premier League title-winning squad and would love to add the World Cup winner to his side.
The Germany international has won just about everything there is to win with the Bundesliga champions and, according to reports in Spain, he is now looking for a move to the Champions league holders.
It remains to be seen if Los Blancos are as keen on the 28-year-old as he is on them, with reports claiming they would prefer to make a move for his team-mate David Alaba instead.
If his move to Spain fails to materialise then a move to Chelsea is said to be next on the list, with Conte looking to add to his defence ahead of their return to the Champions League.
Boateng left Manchester City for Bayern in 2011 for around £11m.