On Tuesday, Messi sent a fax to Barca officials indicating his desire to leave the club, requesting that he be let out of his contract.
The Argentine is unhappy at the decline of the Blaugrana on the field, as in 2019/20 they failed to win any major silverware for the first time in 12 years.
He is also disillusioned with how the club is being run at boardroom level.
It was actually Bayern who arguably caused the final straw for Messi as it was the German side's 8-2 destruction of Barca in the Champions League which led to Quique Setien being fired and then the player handing in his request to leave.
Only a handful of clubs around Europe could potentially afford a play for Messi's services, on wages alone, but Bayern are not one of them.
“No,” Rummenigge emphatically responded to Tuttosport when asked if Bayern could potentially sign Messi.
“We can’t pay a player of that size. It is not part of our policy and philosophy.
“Honestly, hearing that Messi could leave Barcelona makes me a little sad. Leo has written the history of that club and in my opinion he should end his career with the Blaugrana," he added.
The clubs thought to be leading the race to sign the six-time Ballon d'Or winner are Inter and Manchester City.