Boateng, 29, is widely considered as one of the finest defenders in Europe but has seen his career stall over the past two seasons due to injury.
And after seven seasons at the Allianz Arena, the 2014 World Cup winner is seeking a new challenge, according to the Bundesliga champions.
Paris Saint-Germain and Manchester United were subsequently offered the chance to sign the Germany international.
However, United passed up the opportunity due to his checkered injury history before their transfer deadline shut two weeks ago.
As a result, the Bavarian giants admit his exit is currently in limbo.
"Jerome has told [chairman] Karl-Heinz Rummenigge he wants a change," President Uli Hoeness told Sky News.
"It's 50-50. We'll be happy if he stays, he is a fantastic player. [If he goes] I believe it will be Paris.
"When a player wants to leave and we have alternatives, then we busy ourselves with that. A decision will be reached in the next few weeks."