Luhrmann (The Great Gatsby, Moulin Rouge) penned the script with Craig Pearce. The biopic will chronicle Presley's rise to stardom and his relationship with Parker.
Luhrmann is expected to decide who will portray Elvis in the coming weeks.
Styles, a former member of One Direction, made his acting debut in Christopher Nolan's Dunkirk. Elgort, of Baby Driver fame, is set to star in Steven Spielberg's upcoming adaptation of West Side Story.
Teller will be seen in Top Gun: Maverick and Butler has a role in Quentin Tarantino's Once Upon a Time in Hollywood, which arrives in theaters July 26. Taylor-Johnson is best known for his roles in Kick-Ass and Avengers: Age of Ultron.