"It's right that we are interested in signing Arjen Robben," said general manager Uli Hoeness on Bayern's website.
"He's on his way to Munich. He's one of the best attacking players in Europe. He's always been on our wish list," added Hoeness.
He said Robben could even make his Bundesliga debut in Saturday's match at home to champions VfL Wolfsburg, when Bayern will try to pick up their first league win of the season.
The Bavarians have taken only two points from three games in their worst start for 43 years.
"If everything goes well, he can be on the transfer list tomorrow," said Hoeness.
"In theory, it's then possible for him to play on Saturday. But he must pass all the (medical) tests first."
Spanish media reported the German club had agreed to pay 25 million euros for the 25-year-old.
Robben told reporters at Madrid airport on Thursday he did not want to leave Real but the club wanted to sell him.
"There have been a lot of lies but in the end I have to decide what is best for me because I want to play and show what kind of footballer I am," he said.
"Munich has given me a very good feeling."
Robben made his debut for Real in September 2007 after moving from Chelsea but despite some scintillating performances last season his career has continued to be blighted by injury.
The arrival of Cristiano Ronaldo and Kaka at the Bernabeu has made it less likely he can command a regular place in Manuel Pellegrini's team this season.
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