The 25-year-old has been linked to a Bayern team which has so far struggled to their worst Bundesliga start in 43 years having taken just two points from their first three games and were humbled 2-1 by minnows Mainz on Saturday.
Real's president Florentino Perez confirmed the deal on Spanish radio on Thursday and the Dutchman looks set to join Munich in a deal worth €25 million ($36 million).
But Bayern spokesman Markus Hörwich refused to confirm the transfer and said: "Transfers to Bayern Munich are only confirmed by us and we announce things only after we have done our homework."
Bayern team manager Uli Höneß has hinted there may be more transfers before the deadline on August 31 while Robben looks set to struggle for first team football in Madrid after Real signed Cristiano Ronaldo in the summer.
And Bayern's director of sport Christian Nerlinger told the Munich Merkur: "We are not looking for a right-sided defender or a goalkeeper and we will not rush into buying someone."
Bayern got back to winning ways after Saturday's horror result in Mainz with a 3-1 win over Berlin's second division side FC Union as goals by Ivica Olic, Brazilian Breno and defender Edson Braafheid all scored.
Robben's arrival would complement Bayern's playmaker Franck Ribery and give the German giants more options in attack, but their Dutch coach Louis van Gaal says he is letting Munich's bosses Höneß and Karl-Heinze Rummenigge look after any transfers.
"I still have no influence on who Bayern buy and I have not yet got the team to buy into my philosophy," said the Dutchman.