The 'Green Zone' director - who was recently rumoured to be hot a possibility for the head role in the forthcoming 'Cleopatra' movie starring Angelina Jolie - is said to have written a script entitled 'Memphis' about the events leading up to the civil rights campaigner's last moments before his assassination on April 4, 1968.
According to Vulture blog, the movie would be a "human portrayal" of King's last days, taking a look at his personal life, including a faltering marriage, cigarette and alcohol problem and weight gain.
He had also lost the support of President Johnson because of his outspoken views on the Vietnam war.
Scott Rudin is in preliminary talks to produce the movie with Greengrass, although there is no word about who would portray the legendary lead figure.
Discussing the possibility of a 'Cleopatra' project headed by Greengrass, Rudin recently said movie bosses "like the idea" of him working on it, although Angelina Jolie has yet to be approached about such a move.