The German winger has made it clear he will not sign a new deal with City.
He would like to leave for his home country, and Bayern is the club of his choice for next season.
According to Bild's Christian Falk, the first Bayern bid for Sane is €40 million.
It falls well short of what City want for Sane, who has a deal until 2021.
The Germans will have to offer more, or risk waiting for Sane until next summer.
City manager Pep Guardiola acknowledged the player would like to leave, but City sources in the British press have indicated they want a huge fee for one of their most talented youngsters.