The Borussia Dortmund number nine has agreed to a move to the UEFA Champions League title holders, but Lewandowski's manager Maik Barthel said the Bernabeu club's attempt to sign his player was an audacious one.
"There was an immoral offer from Real Madrid," Barthel told Spiegel Online.
"We listened to their offer out of respect for such a big club, but breaking our word never crossed our mind."
Barthel claimed Real were prepared to pay the 25-year-old €9.5million a year, and a €10m signing bonus.
Lewandowski has signed a five-year deal with Bayern, commencing on July 1, and penned a later to Dortmund fans via the club's website on Monday.
"Dear BVB fans, I would like to say a few words to you," the letter read.
"With half a year to go, there are still some goals we want to achieve together.
"I hope you will support me, even if some of you don't agree with my decision! I for one will give my all for BVB!
"Yours, Robert Lewandowski."
