Cooper earned his nod for helming A Star is Born, Cuaron for Roma, Farrelly for Green Book, Lee for BlacKkKlansman and McKay for Vice.
Cooper also received a mention for Best First Feature Film. He will compete in the category with Bo Burnham who directed Eighth Grade, Carlos Lopez Estrada who made Blindspotting, Matthew Heineman who lensed A Private War and Boots Riley who helmed Sorry to Bother You.
Cuaron was named Best Director at Sunday's Golden Globe Awards ceremony where Roma won the Best Foreign Language Film honor, Green Book scored the prize for Best Comedy or Musical Film and Bohemian Rhapsody earned the accolade for Best Drama Film.
The nominees in the DGA race for Best Director of a TV Drama are Jason Bateman for Ozark, Lesli Linka Glatter for Homeland, Chris Long for The Americans, Adam McKay for Succession and Dana Reid for The Handmaid's Tale.
The Americans won the Golden Globe for Best TV Drama and The Kominsky Method garnered the award for Best TV Comedy.
Shortlisted for the DGA Award for Best Director of a TV Comedy are Donald Glover for Atlanta, Bill Hader for Barry, Hiro Murai for Atlanta, Daniel Palladino for The Marvelous Mrs. Maisel and Amy Sherman-Palladino for The Marvelous Mrs. Maisel.
Winners of the DGA Awards are to be announced Feb. 2 at the Hollywood and Highland Center's Ray Dolby Ballroom in Hollywood.