The experienced midfielder has stressed that his transfer is not related to the signing of the Germany international, insisting it was his decision to depart the Santiago Bernabeu
The former Spain international had lost his starting berth at Madrid following the signing of Kroos and James Rodriguez, yet Alonso has stressed that he was not afraid of the competition for places.
"The club did not want me to leave. It was my own decision. After winning La Decima, I felt I needed a new challenge. I felt that the time had come to go," Alonso said at a press conference.
"I did not decide to leave because of the arrival of Kroos. Madrid have made a great signing with him. We would both have played a lot.
"I am not upset or anything. Signing Kroos was a well-thought decision that I respect. Madrid have a great team and can fight for all titles.
"I need a certain motivation to get the best out of myself. Winning La Decima was that motivation at Madrid, but I needed a change to reach my peak again. The time was right to move on.
"Bayern were the best option for me. We will be challenging for all titles and I cannot wait to get started. Bayern are on par with Madrid."
The 32-year-old has signed a two-year deal with Bayern.