The Welsh wizard clinched his third winners’ medal in four years in his hometown of Cardiff on Saturday as Real beat Juventus 4-1.
Afterwards Bale said he wants to win more with Los Blancos, who are starting to emulate the great side of the 1950s.
Bale played the final 13 minutes of the final having only trained for five days following a six-week lay-off with a calf injury.
But he was all smiles after lifting the Champions League trophy yet again, having collected a LaLiga winners’ medal last month.
However, attention quickly turned to the 27-year-old’s future a season of frustration at the Bernabéu.
Asked if he would remain with Real next season, Bale replied: “Yes. I have signed a long-term contract at Madrid. My family is happy and I am happy so we will continue what we are doing.”
Even if clubs in the Premier League want to see him again? “As I have said, I am happy here at Madrid. We are winning trophies and I am happy.”
And despite being forced to start on the bench at the Millennium Stadium, Bale was delighted to help Real clinch their first league and European Cup double since 1958.
“It was a very special occasion for me personally,” he said. “It was an incredible feeling to lift that trophy here.
“I only found out I wasn’t going to start just before the game but I always knew really because I’d only been training for five days with the team and I was really lucky to be involved, to be honest.
“I’ve worked double sessions for three or four weeks to get myself ready for this, to get myself fit and recover from the surgery – which still has a little more healing to do – but I will be able to rest in the summer, do some more rehab and then come back next season stronger.
“It’s three (Champions League) trophies in four years, not many people can say that. It’s great to be doing that and I want it to continue. It’s why you’re a footballer, it’s the reason came here.”
He believes Zinedine Zidane’s squad can make the same sort of impact that the great side of the 1950s did when Alfredo di Stefano and Ferenc Puskás led Real to the win first five European Cups.
“I don’t see why not,” Bale replied when asked if the current side could replicate that iconic side. “We have won three Champions League titles in four seasons so we have to be in that category.
“All we can keep doing is keep working hard and hopefully there will be more to come. I think we can get better. We are still quite young as a whole team and we have a great squad.
“We haven’t just got a great XI, we have a great squad so if any injuries come along there are other people who can do a job. We are strong at the moment. We are confident and the plan is to win more.”
And he says Real Madrid are getting into the winning habit and still get the same buzz from lifting world football’s biggest club prize.
“It’s a little bit more relaxed as we get used to it but it’s great, a really good atmosphere afterwards. I think it adds to the history of this club.
“This club is the biggest in Champions League history ever and now we have even more. Its great to be a part of and we want more.”