The Arsenal midfielders decade-long stint in north London is set to come to and end in June when his current contract expires.
Ramsey was close to penning a new deal last month but the offer was withdrawn on the advice of Unai Emery, who would reportedly prefer to use the funds available to him to strengthen other areas of his squad, rather than reward the 27-year-old with a significant pay rise.
The Wales internationals situation has alerted the likes of Liverpool, United and Chelsea, while Serie A giants Juventus and AC Milan are also keeping close tabs.
According the Express, however, Bayern are keen to make the first move and with Ramsey able to speak to foreign clubs when the transfer window reopens in January, the Bundesliga giants are at a significant advantage.
Ramsey has struggled to establish himself in Unai Emerys line-up this season and again started on the bench during Saturdays 1-1 draw against Liverpool.
Speaking about Ramseys mindset ahead of yesterdays match, Emery said: The most important thing for me with Aaron is, I spoke with him last week, his spirit, his quality, his confidence, his focus to us in training, in matches like today.
He played for 20 minutes against Crystal Palace, he played with this spirit.
Hes focused with us, with the training and in every match like today also. He is working very well.
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