The left-back, 27, joined Chelsea from Brighton in 2022 after coming through La Liga champions Barcelona's academy and moving to Getafe.
Real Madrid president Florentino Perez appointed Jose Mourinho as coach this week after he was reelected earlier in June, following two seasons without winning a major trophy.
Los Blancos are set to sign Liverpool defender Ibrahima Konate at the end of his contract at Anfield and have been linked with Bernardo Silva and Inter Milan's Denzel Dumfries.
Chelsea finished 10th in the Premier League last season in a disappointing campaign, with Cucurella critical of the decision to sack Enzo Maresca in January.
The defender had two years left on his deal at Stamford Bridge, where he won the Club World Cup and UEFA Conference League in 2025.