The Daily Star Sunday says City have been told £25million-valued Muller could quit Bayern as part of a shake-up at the Allianz Arena.
And the Munich club, looking to rebuild after failing to win a trophy last season, want City striker Edin Dzeko.
Muller, 22, Germany's attacking midfielder or striker, is believed to be highly regarded by City boss Roberto Mancini.
But City could face competition from Chelsea, Arsenal, neighbours Manchester United and Real Madrid. However, the key to the deal could be Dzeko, who is expected to leave the Premier League champions this summer.