Ramsey is set to leave the Gunners at the end of the season when his contract expires after the club withdrew an offer of a new deal.
According to Neil Fissler of the Express, Bayern are interested in signing him and will look to steal a march on United and Liverpool by approaching him in January.
Premier League teams will have to wait until the summer to do so, but foreign clubs are free to negotiate a contract with him at the beginning of 2019.
The Welshman has achieved much during his time at the Emirates Stadium.
Ramsey joined the Gunners in 2008 and overcame a broken leg early in his career to become an important player for Arsenal.
The 27-year-old is a dynamic and versatile operator in midfield whose box-to-box drives help the Gunners advance up the pitch.
Upon reaching the final third, Ramsey is one of the most productive members of Arsenal's team.
His ability to score and create goals makes him a valuable asset, so it's of little surprise clubs of United and Liverpool's calibre are interested in snapping him up.
Indeed, as a free transfer, he could prove to be an absolute bargain.
He'd be well-suited to the intensity of German football, and Bayern's ability to deliver silverware and compete at the business end of the UEFA Champions League will make them a tempting proposition for Ramsey.
However, they also have a great deal of midfield talent available to them already, so if he did join, he may not play as significant a role as he might like.