Alvarez is now starring with Atletico in LaLiga, having made the move 18 months ago from Manchester City.
He told L'Equipe: "I had the opportunity to come to Madrid when I was 11. My father accompanied me to Spain for about 20 days.
"I trained with Real Madrid and then took part in a tournament in Peralada that we won. But for me to stay, my whole family would have had to come and live with me. It was a wonderful experience, but it was still too early for me."
Asked about replacing Robert Lewandowski at Barcelona next summer, Alvarez added: "Frankly, I don't know. I see what's being said on social media... There's a lot of talk in Spain about me and Barcelona. When I signed for Atlético last year, there was also a lot of talk about Paris.
"It's true that there were discussions between the PSG management and my agent; they showed interest in me, but nothing materialized. For now, I'm focused on Atlético. We'll see what happens at the end of the season."