Bayern Munich could have sold Thomas Muller to Manchester United for a world-record fee in 2015, according to Karl-Heinz Rummenigge.

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.

"I hope so [that the fee rises], because then he would be a really good signing," he added.

"If you want to have a player with world-class potential, then you have to act quickly, much earlier than had previously been the case, otherwise these players are gone."

Rummenigge also revealed Bayern had the opportunity to sign Kevin De Bruyne last year.

The Belgium international, who moved to Manchester City, was considered too much of a risk by the Bavarian giants.

"We had an offer to sign De Bruyne in 2015 for the following year," Rummenigge added.

"The fee plus salary amounted to over 100 million euros, so the risk was too high.

"We have now signed a centre-back in Mats Hummels, because we have enough quality and quantity in attack."