Isco impressed as a substitute in a tense final at San Siro, which Madrid eventually won on penalties after goals from Sergio Ramos and Yannick Carrasco ensured the match finished 1-1 after extra-time.
The Spain international fell out of favour under former coach Rafael Benitez and came close to leaving Santiago Bernabeu in the mid-season transfer window, but he has enjoyed a more sustained first-team role since Zinedine Zidane took charge in January.
And the former Malaga man says he is keen to enjoy more success in the capital as long as the club want him to stay.
"I am very proud of my team. We had a difficult season, but we have never given up, so we deserve it," he said after the match.
"I hope to stay. If the club wants me here, I will stay. I am very happy here and I hope to keep winning trophies for this club. Zidane has done an amazing job with us. "It is in our DNA to fight to the very last minute. We showed that in La Liga and right now in the Champions League. We have played against a very competitive opponent who made things very difficult, but thank God we can bring the trophy to the Bernabeu."