According to Deadline, Green is being courted for one of the three main adult roles in the film as production nears.
Green has worked with Burton before, on Miss Peregrine's Home For Peculiar Children and Dark Shadows.
Dumbo is said to be a top priority for Disney, which has seen success with other live-action adaptations, including Alice in Wonderland, Maleficent, Cinderella, The Jungle Book and the upcoming Beauty and the Beast.
Originally an animated film that hit theaters in 1941, Dumbo follows the story of a baby circus elephant who uses his large ears to fly.
The script was written by Ehren Kruger, who wrote the screenplays for The Ring, Scream 3 and three of the films in the Transformers series.