Asensio, who is on duty with La Roja at the World Cup, is set to become a free agent when his existing deal with Los Blancos expires at the end of June.
Repeatedly linked with a move away from the Santiago Bernabeu, with Barcelona reported to be among his possible destinations, the forward has started just once in LaLiga for Madrid this season.
Nevertheless, the 26-year-old hopes he can prolong his stay with the reigning Champions League winners, for whom he has made 250 appearances since arriving from Mallorca in December 2014.
"There is talk that I am changing Real Madrid's mind. But I have always tried to give the maximum that I have inside, and then it will be known," he said.
"The one that decides, too, is the club, it not only depends on me. After the World Cup, we will both make a decision.
"What if I want to renew? Hopefully, it can happen, and I will be very happy at Real Madrid for many years. Hopefully, everything will be given to stay 10 more years."
Asensio also offered his commiserations to team-mate Karim Benzema, who was ruled out of the World Cup after failing to recover from a quadriceps problem in his left thigh.
The France international became the first reigning Ballon d'Or winner to miss the finals since Allan Simonsen's Denmark failed to qualify for the 1978 tournament.
"I send him all my encouragement and support from here," Asensio added. "These are things that happen in football, and it has weighed on him not being able to reach 100 per cent. He was very excited about this World Cup."