The 50-year-old has previously won the tournament with Porto and Inter Milan.
However, ahead of his side's last-16 meeting against Manchester United at the Bernabeu, Mourinho has admitted that both he and his club are eager to emerge victorious at Wembley this May.
"There are great clubs in the world who have never won this competition," he told reporters.
"Real Madrid have won nine, I have won two. Real Madrid wants to win its 10th. I want my third.
"If it comes this year good. If not, I will try again next year. I don't think I will end my career with only two titles."
Madrid last won the competition back in 2002.