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Norwich finished the Premier League season in 20th place and will likely be keen to offload any players of value as they take the plunge into the Championship.
The report says Ajax right-back Sergino Dest is Bayern's priority as they look to fill a spot in their squad, but Aarons has emerged as an alternative.
Aarons has a contract with the Canaries until 2024, but he would likely cost around €20m instead of €30m due to Norwich's relegation to the second tier.
Another Bundesliga club, Bayer Leverkusen, are also reportedly interested in securing the England U21 international's services, but he is said to prefer a move to Bayern.
Should Aarons join the Bundesliga champions this summer he would likely be back-up to World Cup winner Benjamin Pavard.
View from Eurosport Germany: 'Why not?'
Eurosport Germany's Marc Hlusiak
Bayern are in need for a right back like Aarons. Still the number one target is Sergino Dest from Ajax, but if that doesn't work out - why not?
Aarons would be a pretty good fit, as he is faster and more offensive-thinking than Pavard. In the future, Pavard should play centre back, so the position on the right would be free. It could be a good fit, but Bayern first will go all in for Dest.