The German giants have announced that former Chelsea wideman Robben will undergo a medical this morning before signing a four-year contract.
Bayern's general manager Uli Hoeness confirmed that a fee, reported to be in the region of 25 million euros, has been agreed between the two clubs and only a few minor details needed to be ironed out in the morning.
"He is on the way to Munich," confirmed Hoeness. "It is right that we are interested in signing Arjen Robben.
"We have already reached an agreement with Real.
"He is one of the best attacking midfielders around in Europe and he has always been the player we wanted."
Robben could be included in coach Louis van Gaal's squad to face Wolfsburg at the weekend and may make an immediate debut.
Hoeness said: "It is theoretically possible that he will play on Saturday. Our intention is for him to play on the right wing with (Franck) Ribery on the left, making our team even more attractive."