This is the first time since 1958 that Real Madrid has won the La Liga and UCL double, but Bale believes they can get better.
Los Blancos eased to the Champions League win, after defeating Juventus 4-1 at Cardiff on Saturday.
Bale sees no reason why Madrid cannot win more major trophies that will put them at a higher pedestal.
The Wales international told reporters as Madrid celebrated their latest Champions League triumph: “I don’t see why we can’t be in that company. We have won the Champions League three times 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.”
He added: “We can get even 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 very confident and the plan is to win more.”