The Bundesliga giants have been heavily linked with a move for the Tottenham forward with the 29-year-old now in the final year of his contract in north London.
talkSPORT understands that Spurs are prepared to reject any offers from Bayern Munich this summer, but according to the former Bayern and Manchester United star, Kane should seriously consider a switch to Bavaria for the benefit of both his professional and personal life.
Asked if Kane could be the missing piece for Bayern, Schweinsteiger told The Sports Breakfast: "Yes, 100 per cent.
"Bayern Munich are looking for a striker, a proper striker like [Robert] Lewandowski.
"Choupo-Moting is not good enough for Bayern Munich and so he's more the plan B solution, but Bayern Munich would of course like to sign Harry Kane."
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Schweinsteiger added: "I think that it would be a great fit for him as well.
"He has to ask himself if he wants to win a league, a title in his life as a football player - you know, a championship or a Champions League.
"I know he's a person who likes to have his family and his friends around him and Munich is the best place for that. You have a beautiful area around where you can live and where your family can grow up and it's safe.
"Also the experience abroad that I had when I went to Man United helped me a lot and I grew as a person as well and of course as a football player he'd have a very good team around him and a very good coach, so there's a lot of things are speaking for him in his favour."
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Bayern boss Thomas Tuchel reportedly angered Spurs chairman Daniel Levy by holding personal talks with Kane in recent weeks.
Levy is known to be a fierce negotiator and Schweinsteiger insists Bayern will not be sucked into paying over the odds.
Revealing a possible obstacle for the deal, he said: "I also know that Bayern Munich are not a club who will pay as much money as maybe the owner of Tottenham is asking for.
"So there will be one guy who puts the first offer on the table and says 'no, we're not paying so much' but they're looking for a striker, that's for sure."