They describe Affleck as the only director who has seen Will Beall's script for Justice League--which is interesting, because that implies that the script is finished.
The actor-turned filmmaker is reportedly Warner's favorite following news that The Dark Knight Rises' Christopher Nolan is unwilling to consider the film. Affleck directed the John Hamm-starring heist movie The Town for Warner Brothers, which did good money and earned the director accolades and award nominations.
The magazine reports that while Affleck may be hesitant to direct the superhero juggernaut (which would essentially take away the possibility of him directing anything else in the next couple of years), he could accept the job and then also take the lead role in Replay, Greg Berlanti's long-in-development science fiction film in which Warner has a substantial stake and for which Affleck was already considering a role.
The article also speculates that if Affleck were to direct the film, he may take a role of his own--while Variety implies that would be a superhero role, it seems unlikely that someone so committed to his directorial career would accept an acting-only role that would eat so much of his time.
His next directorial film will be Argo, a true story of espionage in which the Canadian government helped to rescue some American refugees stuck in the Middle East by working with the CIA--and legendary comic book creator Jack Kirby--to create a fake science fiction movie.