Bellingham has scored 21 goals for Real since signing in the close season from Borussia Dortmund for 103 million euros (£88.5m).
The Englishman scored in added time to secure a 3-2 comeback win over Barcelona on Sunday as Real opened up an 11-point lead over their rivals.
"It’s definitely something I want to do," Bellingham said of his trophy targets. "We’ve still got a few more games to wrap up La Liga and then we’ve got two very difficult games against (Bayern) Munich (in the Champions League semi-finals).
"For the Euros hopefully we can take our form from the last few tournaments and do one better."
Bellingham's impact at Real has been undeniable. His leadership qualities have been praised by his teammates and he is the league's second-highest goal scorer with 17, one behind Girona's Artem Dovbyk.
Having made his debut in 2020, Bellingham has 29 caps for England and scored three goals. England are grouped with Slovenia, Denmark and Serbia at Euro 2024, which begins on June 14th.
