The Arsenal midfielder, 27, is expected to leave at the end of the season when his contract expires after 10 years in North London.
But the Welshman will be free to sign a pre-contract agreement with a foreign club in January.
According to Sportsmail and cited by Daily Mail, Bayern are among the clubs scrambling for his signature.
Niko Kovac is keen to bolster his midfield and Ramsey would arrive in Germany on a free transfer.
The Bundesliga giants will face competition from Juventus, who are reportedly leading the race to sign him.
Ramsey has been with the Gunners since 2008, but they withdrew their new contract offer in September.
And earlier this month it was reported that Real Madrid had been in contact with the 27-year-old in an attempt to lure him to Spain.
He was a pivotal figure in Arsenal's squad under Arsene Wenger, but new boss Unai Emery does not share the same view.
Fans have called for Ramsey to be offered a new deal at the Emirates following his impressive spell of form this season.
But the Gunners boss has confirmed that he will not change his mind.
He said: "That situation was closed two months ago."
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