Muller was repeatedly linked with a switch to United last year, and the Bayern chairman admitted he was the subject of a "mad" transfer bid from the Premier League side.
But Rummenigge cited the importance of having home-grown players in the squad as a reason for passing up the offer.
"Thomas Muller came out of it [the youth academy]. [He is] born near Munich, formed here, the hero of the fans," the Bayern chief told Frankfurter Allgemeine Zeitung.
"We had a mad offer from Manchester United last year, which was not a topic for one second.
"We could have secured a world-record transfer fee."
In May, Bayern announced the capture of Benfica youngster Renato Sanches for an initial €35million.
Add-ons could see the fee for the Portugal international rise to €80m, but Rummenigge is not worried about the potential future expense as it would prove the signing was worthwhile.
