Reguilon, 23, came through the youth set-up at Real and made his first-team debut in October 2018.
The left-back joins for an undisclosed fee but with a reported £27.5m buy-back clause.
"Bale has been an idol for me since that game against Inter," he said.
"Since that game in the Champions League I started to follow him, to watch YouTube videos of him and I became a big fan. I bought his shirts, his books - everything about him.
"And now being able to be here with him, in the club that raised him… I showed him on the plane the tweets that I wrote to him, and the pictures that I have from him, and he enjoyed that."