The German international made the comments from Germany's training camp in Winston-Salem, in the United States, ahead of the national team's World Cup campaign.
"And as for Mourinho, I can't say too much, but I can say it makes me very excited. It's finally going to happen that I play under him. In the past there were many conversations with him, but for some reason it never happened. That's why I'm even happier now to welcome him with us."
Rudiger also spoke of his pride at extending his stay at the Bernabeu after a difficult campaign affected by injury.
"I had a difficult year last year, especially physically. I had to fight a lot with injuries and I managed to come back. Madrid is practically my home. I feel very good, my family feels very good there, and that's why I never really thought about anything else. I just wanted to get back in shape and earn my new contract. And I achieved it."
The defender explained that a move to Real Madrid had never been a long-held ambition of his.
"For me, the Premier League told me 'that's right for me'. And when I won the Champions League with Chelsea, I knew that dream could also come true."
Asked whether his winning mentality from Real Madrid could transfer to the national team, Rudiger reflected on a difficult recent period at club level.
"The last two years at Real Madrid were hard. We weren't successful. But if you look at the profile of players we have there, exactly that type of footballer like Kylian Mbappe, Jude Bellingham or Vinicius Junior are the ones who appear when you need those decisive players in the moments you need them."
He pointed to France's recent World Cup win as an example of that same quality showing up under pressure.
"This Tuesday, for example, France didn't have their best day, but they still came through with two goals. And that's quality. You don't see that very often."
Rudiger was a substitute in Germany's opening match of the tournament, a role he accepted without complaint given the team's collective objective.
"For a footballer it's not easy to be on the bench, but with the national team it's different than at club level. We all have a single objective here and we're all important."
He also addressed the disappointment of Lennart Karl's injury absence from the tournament.
"I can identify very well with him, because exactly the same thing happened to me. A torn cruciate ligament before a European Championship. And I think that has nothing to do with experience or anything like that, it's simply something human. At that moment, the boy is 18 years old, it would have been his first World Cup and for me he is an enormous talent."
Rudiger's renewal at Real Madrid had been in doubt for much of last season as he battled fitness problems, making the new deal a significant personal milestone heading into the tournament.
His comments on Mourinho carry added weight, with Rudiger having held discussions with the Portuguese coach before without those talks leading anywhere.