Kepa joined Los Blancos on a season-long loan deal from Chelsea earlier this week.
The 28-year-old will serve as cover at the Santiago Bernabeu for Thibaut Courtois, who is set to miss most of this season with an anterior cruciate ligament injury.
Chelsea's arrangement with Real Madrid means the Spanish goalkeeper will have to return to Stamford Bridge at the end of his temporary stay with Carlo Ancelotti's side.
But Kepa is keen to remain in the Spanish capital at the end of the season.
“Will I stay after the loan? Hopefully,” Kepa said at his Real Madrid unveiling on Tuesday as reported by Marca.
“Today is the first day of my loan. We have time, we’ll see. Hopefully with my performance I can make that happen.”
“This is one of the most important and special days of my sporting career and of my life,” Kepa told the club website.
“I’m very proud to be at Real Madrid. I’ll never be short of hard work, commitment and effort with this jersey both in training and in matches.
“I hope that we can further add to Madrid’s trophy cabinet and enrich the club’s legend. Hala Madrid!”
Kepa has made 163 appearances for Chelsea since joining from Athletic Bilbao in the summer of 2018. He remains the most expensive goalkeeper ever at £71m.
He could make his debut against Almeria in La Liga this weekend.
