Pogba is currently on international duty with France and spoke of his admiration for the Spanish giants at a press conference.
"Like I've always said, Real Madrid is a dream for anyone. It's one of the biggest clubs in the world. There is also [Zinedine] Zidane as a coach and it's a dream for anybody who likes football. For now, I'm at Manchester. We don't know what the future holds. I'm at Manchester and I'm happy."
Pogba's comments are likely to concern Manchester United, as the midfielder has found his best form at the club under caretaker manager Ole Gunnar Solskjaer:
The midfielder struggled under previous boss Jose Mourinho and had a tense relationship with the 56-year-old:
Pogba was left on the bench for Mourinho's final three Premier League games before he was sacked, and Matias Pogba, the midfielder's brother, said he considered quitting the club.
"Did Paul think about leaving because of Mourinho? Of course," he told Telefoot (h/t Metro). "It comes to mind. But today he is where he is. He was under contract so he had to grit his teeth and keep working."
Pogba has helped Manchester United into the quarter-finals of the UEFA Champions League, and they're in with a chance of finishing in the top four of the Premier League under Solskjaer.
The World Cup winner has backed the Norwegian to take permanent charge of the club:
Former Manchester United striker Zlatan Ibrahimovic has said Pogba could become a world-beater.
"I think Paul Pogba has the potential to be the best player in the world in his position," he told Andy Dunn and Matt Lawless at the Mirror. "He is a good guy, he trains hard, he listens, he wants to become better, he wants to win. He wants to perform in every game."
Meanwhile, Real Madrid are set for a summer overhaul after a disappointing season. The club were knocked out of the Champions League by Ajax at the last-16 stage and lie 12 points behind leaders Barcelona in La Liga.
The club have now brought back Zinedine Zidane as manager, and he has reportedly been promised £300 million to spend in the summer, per Miguel Delaney at The Independent.
Pogba's form this season and his latest comments could attract interest from Real Madrid, although Manchester United will certainly not allow one of their key men to leave without a fight.