Álvarez, 26, triggered a storm in Madrid when he told ESPN after Argentina's 2-0 victory over Austria that "the best thing for everyone is a transfer — I want to fulfil my dream." His comments were understood to be a direct signal of his desire to join Barcelona, a club he has been linked with throughout the season.
The Unión Internacional de Peñas del Atlético responded with a sharply worded communiqué published on social media.
"Julián is proving to be a spoiled brat with less capacity for reflection than a cobblestone. And a traitor at the same level as Courtois or Hugo Sánchez," the statement began.
"Now he must be treated as he deserves: either he pays his release clause or he goes to the stands. We hope Apollo Global puts him in his place, or we will feel doubly betrayed."
The reference to Apollo Global concerns the private equity firm that holds a stake in Atletico Madrid's parent group. Thibaut Courtois and Hugo Sánchez are regarded among the most controversial departures in the club's history.
Álvarez has a contract with Atlético until June 2030 with a release clause set at €500m.
The club have stated repeatedly that they do not regard him as available and will not entertain offers below that figure. Atlético have also confirmed they will file a complaint with FIFA against Barcelona for alleged illegal contact with a contracted player.