The France international has been linked with a move to Manchester United to reunite with Jose Mourinho, but maintains he is happy at the Bernabeu.
"I am happy here and I hope to be here for many years," said Benzema, who earned his second Champions League crown on Saturday as Real ousted Atletico Madrid 5-3 on penalties after a 1-1 draw.
"I thank the people in Madrid for being there for me and my family. Thanks to everybody because they are part of my life and we are going to be used to winning titles."
Benzema added: "I am very proud of the team and I am glad because it is a very important title for everybody.
"I have to thank to the supporters, the club's president [Florentino Perez], the coach [Zinedine Zidane], the players and today we are in Madrid's history."