The new pals recalled the start of their friendship in an interview on Tuesday's (January 14) episode of "The Ellen DeGeneres Show", on which Downey Jr. served as guest host.
"I was such a fan of (Malek's breakthrough TV show) 'Mr. Robot', I basically emailed you, and I was serial texting you and I was trying to FaceTime you and you didn't let me weird you out, and now we're friends," Downey Jr. smiled.
"I couldn't even believe it was you," Malek revealed. "For a second, I got the email from Robert and I thought, 'OK, somebody's playing tricks on me.' And then they kept coming and they were so witty...".
"It got to the point where I thought it could be no one else but you," Malek explained.
And the 38-year-old insisted there was no hesitation when he was offered the job: "Of course, this usually goes through a number of agents and whatnot, but when RDJ calls you, it's like, 'Yeah, sure, I'll come do it'."
Malek voices a gorilla named Chee-Chee in the new film, based on author Hugh Lofting's beloved children's character Doctor Dolittle, with Downey Jr. starring as the eccentric titular character.
Malek went on to admit getting to work with Downey Jr. was a real treat for him, having admired his career for decades.
"I have a big crush on you," he shared. "Thank you for everything you've done for film and cinema. All of these roles from Chaplin to Dolittle now, Iron Man. Everything that you've done has not only been something that we as actors look up to and admire, but you have taken so many of us under your wing and I am so appreciative."
"Dolittle", which opens in the U.S. this week (January 17), also features Emma Thompson, Marion Cotillard, Ralph Fiennes, Tom Holland, and Kumail Nanjiani, as well as the voice talents of Octavia Spencer, John Cena, and Selena Gomez.