Ribery has said he turned down a move to Chelsea as a Bayern.
"We received an offer for Franck Ribéry from England, 85 million euros in transfer fees, plus one player," revealed Rummenigge in an interview with the Munich newspaper, Abendzeitung.
"Uli (Hoeness), Karl (Hopfer) and I discussed it for hours, all three of us were torn. Because 85 million euros would have given us no worries in the future."
Matthijs de Ligt, Lucas Hernández and Robert Lewandowski currently share the record sale price at Bayern for €45m.
Rummenigge continued: "But it was ultimately Karl Hopfer who formulated the decisive sentence that still applies today: 'We don't want to be a sellers' club, just a buyers' club'.
"That was a very important moment for the future of FC Bayern. We kept Ribéry, extended his contract and didn't sell any of our best players after that either."