Real have stumbled in the opening games of their title defence.
But Carvajal said, "My contract runs until next summer. I'm not thinking about quitting right now. I talk about it with my family: I want to enjoy every match, every training session. As the years go by, you realise that you have to give more value to every moment and enjoy it.
"The relationship I have with the club is very transparent, very clear. If both parties agree to continue, I hope to stay here for many years."
On Kylian Mbappé, he also told the Athletic: "It has been an important change after Toni Kroos, Nacho and Joselu left us. With the arrival of Kylian, the coach has to adapt to all the big players we have.
"The other day against Real Betis (2-0 victory), the team showed good signs, we played a fairly complete match and from now on, after the national team break, we will put all the pieces of the puzzle together."
Carvajal does not think Real Madrid will have problems inside the locker room amongst their attacking players.
"It is clear that all three (Mbappé, Rodrygo and Vinícius Júnior ) feel comfortable playing on the left wing - but last year I had a lot of space on the wing and it strengthened my attacking game a lot. This year I don't think there will be any problems.
"When you don't have the ball you have to be structured, but when you have the ball the coach gives us more freedom. The system is up to the coach, he decides."