The Arsenal midfielder hoped to sign a new deal with the north London club but talks came to an impasse this autumn.
Ramsey, currently on £110-a-week sought an improved deal that would reflect his importance to the club.
But The Independent are reporting that Bayern Munich ‘are confident’ of securing his signature.
Mesut Ozil’s weekly wage of £300,000 reportedly frustrated Ramsey, who feels his worth is near that of the German playmaker.
The 27-year-old is the Gunners’ longest-serving player, signing with the club as a teenager in 2008.
The Welshman has won two FA Cups with Arsenal, scoring the winner in two of them.
The club’s inability to pin down their loyal midfielder will see him leave on a free.
Despite no fee, it will prove an expensive deal for Bayern, who are expected to hand Ramsey a lucrative, long-term contract.
The German giants are having their worst season this decade, trailing leaders Borussia Dortmund by seven points.
Niko Kovac, who was appointed as manager in the summer, is under increasing pressure from the board, who are willing to back him in the market.
The club signed midfielder Leon Goretzka on a free transfer in the summer, and are looking to emulate that capture with Ramsey.
Liverpool and Manchester United were also rumoured to be interested in the attacking midfielder, but Jurgen Klopp ruled out the move earlier this month.
United could yet mount a pursuit of Ramsey, but the Welshman did turn down overtures from Old Trafford before signing for Arsenal 10 years ago.
Ramsey has featured in every Arsenal game this season – six as a substitute – scoring one goal and making four assists.
He vouched to see out his contract in north London, making a January exit very unlikely.