Sane suffered a huge setback when he tore his anterior cruciate ligament during City's win over Liverpool in the Community Shield in August, and has been sidelined ever since.
But SportBild says that has not put off Bayern from trying to complete a deal for their long-term target, and they are thought to be continuing to pursue Sane in an attempt to sign him before the end of the transfer window.
Sane is expected to return to action in February, and Bayern think he could give the team a big boost as they look to turn around their so-far disappointing season, with the Bavarians lying in third place in the table.
City manager Pep Guardiola recently refused to speculate about Sane's future, insisting that he is currently only focused on the German international returning to full fitness.
When asked about the interest from Bayern, Guardiola said: "It's a question for Leroy, his agents and the club. I am not involved. What's important now is that he recovers well."