The Chilean international has one year left on his contract at the north London club and has not signed a new deal at the Emirates. Arsene Wenger remains positive that the 28-year-old and Mesut Ozil - who also has one year left - will be at Arsenal next season.
According to the Mirror, the Bundesliga winners will face competition from their former manager Pep Guardiola in signing Sanchez. The Catalan coach wants to sign the forward for Manchester City in the summer.
However, Bayern wants to beat City by offering him a weekly wage of £350,000-a-week ($444,280 per week), which is £50,000 ($63,468) more than what Robert Lewandowski is earning at the Allianz Arena.
Sanchez's agent Fernando Felicevich has already held negotiations with the German outfit. Bayern remain confident of securing the services of Sanchez, who is looking for a move away from Arsenal.
The Gunners are willing to offer him an improved deal that would see him earn £275,000-a-week ($349,076 per week). However, the former Barcelona man is set to reject Arsenal's new deal and the report claims that Bayern will win the race for his signature.
Sanchez is currently on international duty with Chile for the 2017 Confederations Cup in Russia. The player has admitted that his future lies in his representative's hand as he remains focused on the upcoming tournament with his national side.
"I'm looking at what my agent is doing. For now, I'm focused on the [Confederations] Cup in Russia and trying to do well," Sanchez explained.
"The truth is that my agent will see to it. He knows and he will sit down with the club to look for the best option for me. I'd like a lot of things but I'm only thinking about the national team."
Arsenal would be willing to sanction Sanchez's sale to Bayern rather than allow him join their league rivals. The German champions believe a fee of £40m ($50.7m) would be sufficient for them to land the forward.