Pochettino has been out of a job since leaving Spurs in November 2019 after five years in charge.
During his time at Spurs, Pochettino was linked on several occasions to Madrid.
"I don't know if I will ever be coach of Real Madrid because football takes you where it wants to but for sure, it's my dream," Pochettino told Cadena Cope radio.
"If it's not the best, it's among the best clubs in the world.
"It has a great history and I think I'm no different from other football professionals.
"Any coach would have on their dream list to guide Real Madrid."
Pochettino, 46, was linked with Madrid's bitter rivals Barcelona twice this year.
He was reportedly a main contender to replace Ernesto Valverde, who was sacked in January, and he was among the candidates the club considered to succeed Quique Setien, who was shown the door after Barca's 8-2 defeat to Bayern Munich in the Champions League in August before Barca appointed Ronald Koeman last month.
Pochettino, who played and managed Barca's city rivals Espanyol earlier in his career, said he did not receive an offer from Barca this summer.
"I did not see [Barcelona president Josep Maria] Bartomeu but I did have lunch with [Barcelona technical secretary] Ramon Planes," Pochettino said. "We have a great friendship but I was not offered the position to coach Barcelona.
"I'm not anxious about not coaching...perhaps if the virus didn't exist I would have more anxiety to be a part of football but after all that has happened, I'm more relaxed and that has allowed me to get personal and family growth and especially, from a professional level, to develop in other fields."