Carvajal, 32, is widely regarded as one of the best right-backs in world football, and he has played a vital role in the team's recent success, including last season, when he enjoyed one of his strongest campaigns for the club.
The Spaniard has made 421 appearances for Real Madrid, scoring 13 goals and registering 64 assists in the process, winning four La Liga titles, two Copa del Rey crowns and six Champions League trophies in the process.
Carvajal has again been a vital player for Los Blancos early this season, but his contract is due to expire at the end of the campaign, and there have not yet been discussions over a new deal.
The defender has insisted that he is not thinking about retirement and would choose to remain at Real Madrid, but he is open to finishing his career outside of European football.
Carvajal wants to stay at Real Madrid "for many years"
"I have until next summer on my contract. I'm not considering retirement at the moment. I talk about it with my family: I want to enjoy every game, every training session. As the years go by, you realise you have to give more value to every moment and enjoy it," he told The Athletic when questioned on his future.
"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."
When questioned on whether he would represent another European club, Carvajal said: "[USA] It's a possibility. When I decide not to be at Real Madrid, I'm not going to play in Europe and the alternatives are much reduced... it would be one of those three options (USA, Saudi Arabia or Qatar)."
Carvajal also opened up on Real Madrid's early-season problems, insisting that the team will "put all the pieces of the puzzle together" in the coming weeks.
Carvajal insists Real Madrid will find their groove
Los Blancos have beaten Real Valladolid and Real Betis in La Liga this season, but they have also been held to draws by Mallorca and Las Palmas, which has left them second in the table, four points off the leaders Barcelona.
"It's been an important change after Toni Kroos, Nacho and Joselu left us. With the arrival of Kylian [Mbappe], the coach has to adapt to all the great players we have," he added.
"In Mallorca and Las Palmas, we lacked a bit of continuity in our play, but we're going to get there. Against Real Betis, the team showed good signs, we played a pretty complete game and after the international break, we will put all the pieces of the puzzle together."
Mbappe scored twice in Real Madrid's 2-0 success over Real Betis last time out, and the reigning champions will be back in action away to Real Sociedad on September 14.
Carlo Ancelotti's side will then begin the defence of their Champions League title with a clash against Stuttgart in the league stage on September 17.