The Argentina striker admits the courting began during the Paris Olympics.
He told AS: "The first one was, I think, Giuliano (Simeone, Argentina teammate) at the Olympics. It was him, half-seriously, half-jokingly… And it had already appeared on social media as well.
"Like: 'Come to Atlético... Cholo, I don't know... Cholo this... Cholo that...' And then, at that time, Cholo (Diego Simeone, coach) called me and we talked.
"And they started talking to me about Rodri (De Paul), Antoine (Griezmann)... And I was there all day with Giuliano and, little by little, everything started to take shape."
Griezmann a key in the deal
Alvarez admits Antoine Griezmann was a big influence on the move.
He continued: "He was the one who wrote to me... Maybe sent by Cholo (laughs). Being one of the club's legends, having messages from one of the most important people is obviously a big deal for a player.
"He says, 'Come, come with us. They'll treat you well. You'll have a good time. You'll fall in love with the fans...'
"And all these kinds of things added up to make it all come together.
"Yes, he (Griezmann) spoke to me first on Instagram and then, later, on WhatsApp."