Ribery is a known target for big-spending Real Madrid but Rummenigge has revealed three of their European rivals have stolen a march in their pursuit of the 26-year-old.
The former Germany international, speaking in an interview with Bild newspaper, said: "We are relaxed and proud that we have one of the best three players in the world under contract.
"Real Madrid president Florentino Perez and his advisor Pedro Jimenez asked if we were willing to speak with them over a transfer but we rejected the chance and we have no intention of selling Ribery."
Asked if Madrid made an offer, he replied: "No but other clubs have done.
"Chelsea have lodged a bid, as have Manchester United and Barcelona.
"We are completely relaxed about it. The power lies with Bayern Munich."
Bayern have repeatedly insisted Ribery is not for sale, and Rummenigge is adamant the club will not relent and cash in on their prize asset.
He added: "I don't want to talk about sums of money because it is not in our interest.
"The amount is irrelevant. There is always going to be the perception that Bayern Munich are trying to push the price up and that there is a certain value at which we'll have to accept, but I'll say it again: we want to keep Ribery.
"A player has to respect his contract, in his case until 2011. He is on holiday now and then we want to have a successful season with Franck next year."
Rummenigge is concerned at the figures being splashed around this summer with Real Madrid having spent £56 million on Kaka and agreeing a world-record £80m fee with Manchester United for Cristiano Ronaldo.
"It makes me concerned for the future," he said. "These amounts would not be possible in Germany due to the license agreements but in Spain the rules are different.
"I have already arranged a meeting with (Uefa president) Michel Platini to discuss the matter.
"We need to have new rules regarding financial fair play."