The Descendants star is said to be the studio's leading choice to join Andrew Garfield and Emma Stone in the Marc Webb-directed sequel.
Character breakdowns have been sent to talent agencies as the casting process begins, with Electro rumoured to be one of the villains filmmakers are considering for the next instalment, Variety reports.
The studio is also seeking an actor to play Harry Osborn, Peter Parker's best friend-turned-enemy, who was played by James Franco in Sam Raimi's trilogy.
Jeff Pinkner, Alex Kurtzman and Roberto Orci are to write the screenplay for the sequel, which will be produced by Avi Arad and Matt Tolmach.
Production on The Amazing Spider-Man 2 will start early next year with a scheduled release date of May 2, 2014.
Watch an Amazing Spider-Man Andrew Garfield featurette below: