Foster has shown her skills as a director in 1991's Little Man Tate and Home for the Holidays in 1995.
The producers of The Beaver, Steve Golin and Keith Redmon are said to be aiming to begin production of the film in September and New York will be the location.
Gibson has been more involved with work as a director himself in recent years and made the controversial film The Passion of the Christ in 2004, and followed that with Apocalypto in 2006.
Gibson is perhaps best-known for his role in Braveheart, a film which he also directed, as well as the Lethal Weapon series.
