The 25-year-old had been strongly linked with a switch to the Premier League before surprisingly moving to Bayern on Tuesday.
The move will see the Colombian spend the next two years on loan with the German champions, who will then have the chance to sign the attacker on a permanent contract.
Rodriguez has revealed that he is looking forward to "a new challenge", but the former AS Monaco playmaker refused to rule out the possibility of turning out in a Real Madrid shirt once again.
"It is a new life for me. Everything is new, but Munich is a beautiful city. I will try to learn German," Rodriguez told reporters.
"The door to Real Madrid is still open, but for now I only think about the next two years here. After that we'll see and make a decision. It is a new way but no step back. A new challenge."
Rodriguez has scored 36 goals in 110 appearances for Real Madrid following a 2014 move from Monaco. Last term, he scored 11 times in all competitions, but started just 13 of his club's 38 La Liga fixtures.