Real qualified for their fourth consecutive semi-final after surviving a scare to beat Borussia Dortmund 3-2 on aggregate.
The La Liga giants, who were beaten 2-0 in the second leg on Tuesday, will face either defending champions Bayern Munich, Atletico Madrid or Chelsea in the final four.
Speaking to Real fans in a Google 'hangout', the Wales international set his sights on the final at the Estadio da Luz in Lisbon on May 24.
"The next two games are very important," Bale said. "It would be a dream to play in the final."
Despite their run of semi-final appearances, Real have not reached the final of the prestigious tournament since defeating Bayern Leverkusen in the 2002 final.
Bale has made an impressive start to life in Madrid, scoring 18 goals for Carlo Ancelotti's men in his debut season in Spain following his world-record transfer from Tottenham.
But that has not stopped talk of a potential return to the Premier League, with some reports suggesting he could move back to England once his daughter starts school.
The 24-year-old, however, intends to remain at the Bernabeu for years to come.
"I'm having fun playing for Real Madrid, enjoying it a lot.
"I want to stay here a long as possible and win a lot of trophies. Let's hope I can."
