The 24-year-old has continually been linked with a move back to his homeland which has seen City give up almost any hope of negotiating a new contract after nearly two years of talks with the Germany worldwide.
Bayern is set to make an initial payment of 49 million euros and City could receive a further 11 million euros based on Sane's achievements with the German champions, the person told The Associated Press.
Sane suffered a torn ACL in the Community Shield against Liverpool in August of a year ago and did not play again until the 5-0 win against Burnley on June 22nd.
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The 24-year-old was reportedly close to signing for Bayern last year before suffering a knee injury.
City manager Pep Guardiola said last week that Sane, whose present deal was due to expire at the end of next season, had refused the offer of a new contract and wanted to leave.
"We have enough good players up front", Guardiola said. Sane is expected to sign a five-year deal.
"We want players who want to play with this club, we want full commitment to try to achieve our targets. I love him so much, he's incredible, I've nothing against him - he wants another adventure".