However, Germany's 2014 World Cup hero will stay put at the Allianz Arena and fight for his spot in new manager Carlo Ancelotti's first team according to national team boss Low.
"I've been in constant contact with Mario and he's kept me informed all the time," Low said at a press conference on Wednesday.
"He's had a long talk with Ancelotti and I think a few falsities were said about that, but a coach can never give a guarantee and he told Mario that everybody starts from scratch from the start of the season under him.
"Mario has had offers, but I always felt he is in Munich with his heart and he wants to prove himself there.
"He told me he wants to stay in Munich and take his chance with the new coach and I think that's a fine decision because it's what his feeling tells him to do, so it's the right decision for him."
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