The Red Devils were heavily linked with a move for Muller at the start of last season, but no deal materialised and the striker remained in Munich.
Speaking to German newspaper Frankfurter Allgemeine Zeitung, Rummenigge has now claimed United did indeed make a world record offer for the 26-year-old, which the Bavarians turned down without a second thought.
"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," he said.
"Thomas Muller came out of (the youth academy). (He is) born near Munich, formed here, the hero of the fans."
Sticking on the theme of transfers, Rummenigge also stated they had a chance of signing Kevin De Bruyne from Wolfsburg before he joined Manchester City, but said the financial breakdown of the deal meant it was too big of a risk for them to make.
"We had an offer to sign De Bruyne in 2015 for the following year," he 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."