Sanchez, 28, will be out of contract next summer and his long-term future with the Gunners remains up in the air.
But, according to Calciomercato journalist Matteo Pedrosi, he is close to sealing a move to Bayern.
The German giants are said to have beaten Paris St-Germain in their pursuit of the forward.
However they apparently still need to agree a fee with Arsenal.
Sanchez has starred for the north London outfit once again this season, scoring 26 goals across all competitions.
He arrived at the Emirates in a £36m move from Barcelona three years ago and has since made 142 appearances for the club, winning two FA Cups in the process.