Depp has played Captain Jack Sparrow in the previous three films
Rob Marshall, the Oscar-nominated director of Chicago, is in talks to direct the fourth instalment of the Pirates of the Caribbean franchise.
Studio Disney is eager to secure a director, to allow filming to begin next year, says The Hollywood Reporter.
The Ring director Gore Verbinski, who directed the original hit film, and subsequent sequels, announced he was stepping down from the series in April.
The previous three films earned more than $2.6bn (£1.5bn) around the world.
It was confirmed that Depp would reprise the role of Captain Jack Sparrow last year. The actor is also scheduled to play Tonto in another Disney franchise, The Lone Ranger.
Auditions are reportedly under way for the latest Pirates outing. Neither Keira Knightley nor Orlando Bloom are expected to return.
Marshall is currently finishing work on the forthcoming musical Nine - a reworking of Federico Fellini's 81/2 - starring myriad Oscar winners including Daniel Day-Lewis, Judi Dench, Sophia Loren and Nicole Kidman.
Marshall's previous films include 2005's Memoirs of a Geisha and Chicago, which won six Oscars in 2003, including Best Picture.