"Have you seen Messi cleaning the floor with our jersey and flag?" Alvarez wrote on Monday on his Twitter account, which has 2.2 million followers.
"He should ask to God that I don't find him!" Alvarez added in another Tweet accompanied by two fist emojis, an enraged red face and a flame.
The Argentina team did not immediately respond to a request for comment about Alvarez's statement.
Messi, who is playing in his fifth World Cup, scored in Argentina's 2-0 win over Mexico on Saturday as the South American giants got back on track after suffering a shock loss to Saudi Arabia in their opening game.
Mexico must win their final Group C match against Saudi Arabia on Wednesday to have any chance of reaching the last 16. Argentina must beat Poland to be sure of a place in the knockout stages, though a draw would be enough if Mexico and Saudi Arabia also draw.