Madrid will face city rivals Atletico at San Siro on Saturday hoping for a repeat performance of the sides' 2014 Champions League final meeting, which Real won 4-1 after extra time.
Barca recovered from a slump of form in April – a run that began with defeat in the Clasico - to win their final five games of the season and triumph in La Liga by a point from Madrid.
Meanwhile, fielding the ineligible Denis Cheryshev resulted in the Bernabeu club's expulsion from the Copa del Rey, which Barcelona won with an extra-time win over Sevilla last Sunday.
However, Ronaldo believes there is not greater prize in football than the European Cup.
He told La Sexta: "Winning the Champions is so much more than Barcelona's double.
"A Champions League win is a dream come true and I'm telling you that from the heart."
Ronaldo refused to rise to the bait of Gerard Pique, who appeared to mock one of the Real Madrid star's goal celebrations as Barca celebrated their title win.
"Even when they win a title they have to remember Cristiano," he said. "It's incredible that I'm talked about in a good or bad light but I have no personal problem with Pique."
Ronaldo was also in magnanimous mood when discussing Luis Suarez pipping him to the 'Pichichi' - La Liga's top goalscorer award.
The Portugal international added: "Obviously I was annoyed at not winning and was close to doing so. However, full credit must go to Luis Suarez who has had an excellent season."