The 27-year-old is a free agent at the end of the campaign as the Gunners are not looking to offer him a new contract.
The Bundesliga giants would offer him a deal in the winter window, as per the report, and steal a march on the rivals for the Welshman.
The midfielder came on as a substitute in Arsenal’s 1-1 draw with Liverpool on Saturday. Ramsey has played 11 Premier League games for the Gunners this season with one goal and three assists.
Signing him on a free would be a great bit of business, however, Bayern can wrap up a deal for the 27-year-old in January itself.
The Arsenal star can agree a pre-contract with foreign clubs, but hopefully, he will take his time before making a decision.
We could do our bit by offering Arsenal a cut-price on the 27-year-old and ensure he does not leave the Emirates on a free.
United have done business with Arsenal over another star player last season, Alexis Sanchez, and even though that deal hasn’t worked out great for us, there is no doubting that Ramsey would be a decent player to have under Jose Mourinho.