The German Bundesliga champions say that the 24-year-old right-back Odriozola put pen to paper on Wednesday after passing the mandatory medical and started training shortly afterwards.

"I'm ready, I don't want to waste any time. I want to play and know that the expectations here a very high. I'm already thinking about the game against Schalke," he said at his presentation, looking ahead at Munich's next Bundesliga date on Saturday.

Coach Hansi Flick has asked for reinforcements in the January transfer window as Munich aim to make up a four-point deficit on leaders RB Leipzig in the Bundesliga, and also want to go far in the German Cup and the Champions League.

"Alvaro is in top shape, that's very important. He can play and show his qualities straight away," sports director Hasan Salihamidzic said.

Four-times capped Odriozola came to Real in 2018 from Real Sociedad but never managed a proper breakthrough at the Bernabeu because Dani Carvajal is first-choice on the position.

Flick had called for reinforcements because the squad has been hit by a series of injuries, with centre back Niklas Suele still out for a while and Lucas Hernandez, also a centre back, only just having returned to training.

"Following internal discussions, we agreed to follow the wishes of head coach Hansi Flick in strengthening the defence and decided to go for Alvaro Odriozola," Bayern chairman Karl-Heinz Rummenigge said.

"We are delighted about the signing and thank Real Madrid for the friendly and cooperative talks."

It was not clear Wednesday whether the deal includes an option for Bayern to make the deal permanent in summer.

Australian Associated Press