The left-back arrives from Chelsea, where he had established himself as a regular for Spain at left-back since his move to Stamford Bridge from Brighton, and joins a Real Madrid squad already reshaped this summer following Jose Mourinho's appointment as head coach.
Speaking to El Mundo, the defender described how fast the move was completed.
"It all happened in a day and a half, two days," Cucurella said. "For me, that's much better, much quicker, no headaches."
Cucurella confirmed in the interview that there had been other clubs interested, but that everything became clear once Real Madrid entered the picture.
"There were different options, but when the one from Real Madrid arrived, I didn't think twice," he said. "I think it's a unique opportunity. Playing for Real Madrid is an honour, and not many players can say that. It has a mystique. I don't know how to put it. It's unpredictable. I think that's been shown many times in historic comebacks."
The left-back also spoke about a conversation with head coach Jose Mourinho, who had already told him he would be important to the team.
"We were talking and he told me he was really looking forward to working with me, that I would adapt very well, that Madrid was a great club," Cucurella said. "Then he just told me to have a great World Cup and that we'd see each other in Madrid."
Cucurella was also asked whether Mourinho had told him he was the club's only target at left-back.
"No, I don't know if he said that to me," he said. "He told me he wanted me, and that's it, I'm very happy with that confidence. I can't wait to start working alongside him."
The move means Cucurella will link up with several Spain team-mates already at the Bernabeu once his World Cup commitments are finished, joining a squad that has added a string of new faces in quick succession this summer as Mourinho looks to rebuild after a campaign without a major trophy.
Real Madrid have made a habit of announcing new arrivals at a rapid pace in recent weeks, with Cucurella's deal following closely on from other additions to the squad, all completed before the World Cup had even reached its halfway point.
For Cucurella, the quick resolution means he can now focus fully on Spain's World Cup campaign without his club future hanging over him for the rest of the tournament.