Kepa Arrizabalaga has been capped 13 times by Spain
The Spain international's season-long loan to Real was completed on Monday.
Kepa will cover for Real's regular number one Thibaut Courtois, who is set to miss most of the season with an anterior cruciate ligament injury.
"Will I stay after the loan? Hopefully. Today is the first day. We'll see. Hopefully with my performance I can make that happen," Kepa said.
The 28-year-old made 163 appearances for Chelsea having joined from Athletic Bilbao for £71m in August 2018 - the highest fee ever paid for a goalkeeper.
He was facing competition for a starting berth at Stamford Bridge from Robert Sanchez who the Blues signed from Brighton for £25m this summer.
Real won their La Liga opener 2-0 against Athletic Bilbao with Ukrainian goalkeeper Andriy Lunin making a first start since April.
Kepa could be handed his Real debut by coach Carlo Ancelotti in a league fixture away to Almeria on Saturday.
"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," added Kepa.
"I hope that we can further add to Real's trophy cabinet and enrich the club's legend."
-- Courtesy of BBC Sport