The Spanish champions welcome the Red Devils to the Bernabeu on Wednesday for the first leg - and Mourinho hopes the teams rise to the occasion.
The Portuguese, who was at Old Trafford to see United beat Everton 2-0 on Sunday, told MUTV: "It is the match the world is waiting for. People think we are under pressure because a big team will be out.
"But it is the kind of match we want and the people are not waiting for any other game. I hope we give them what they want."
On pitting his wits against old foe Sir Alex Ferguson, the former Chelsea boss added: "I feel privileged about that because he is such an important person in the world of football and, more importantly, he is good person.
"I have always had a fantastic relationship with him and I am proud of it. We have had so many matches between us which started with Porto. We had some with Chelsea, Inter and now Real. Some I had won, some I have lost, others I have drawn.
"Of course, I want to win and he wants to win, but I believe the loser will have a little bit of space to feel a little bit happy because of the friendship. But make no mistake, I want to win."