Bichir sent bottles of high-quality tequila to Brad Pitt, George Clooney, Gary Oldman and Jean Dujardin.
The alcohol also came with a special note from the actor.
"The note said it was an honour to be among them," Bichir told E!. "It is an honour and I was just overwhelmed to be with their names. I wanted to let them know that I admire their work and I've been a big fan forever."
He went on to joke: "I want to get them drunk so they can talk only beautiful things about me."
Bichir, who has been acting in both Mexican and American films since 1977, was also nominated for an award last night (February 25) at the Independent Spirit Award ceremony. He lost out to The Artist's Jean Dujardin.
The actor will compete against Dujardin again at tonight's (February 26) ceremony. He has been nominated for his performance in A Better Life, in which he plays a poor Mexican father, struggling to protect his only son from gang violence and immigration agents.
Watch Demián Bichir in the trailer for A Better Life below: