The Bundesliga giants were thought to be frontrunners for his signature, but they have publicly expressed concerns about alleged wage demands.
Now, Rummenigge has claimed that it would be pointless for Bayern to sign Sanchez as they have already bought three attackers in James Rodriguez, Serge Gnabry and Kingsley Coman, who has joined from Roma on a permanent basis.
"I don't know what [Sanchez] has said but we are no longer involved," the Daily Mail quotes Rummenigge as saying. "We are in agreement with [Carlo Ancelotti] and we won't be doing another deal for an attacker - that makes no sense."
Manchester City, Chelsea and Paris Saint-Germain have also been linked to the Chile international, whose current contract at Arsenal expires at the end of this season.
Sanchez recently hinted that he may leave Arsenal by admitting that he wants to play Champions League football, which will be absent at the Emirates after they finished in a Europa League place.