The move would require multiple new agreements with rights-holders and actors involved in the production.
Talks with lead actors including Martin Freeman, who plays Bilbo, and Ian McKellen, who plays Gandalf, have been taking place in recent weeks, the LA Times reported.
It was not clear how much new shooting would need to take place to add the extra movie or how it would reshape the plots of the first two Hobbit films.
A climactic battle at the end of The Hobbit book may need to held for the third film.
Jackson first raised the idea of making a third Hobbit film in an interview with HitFix at Comic Con.
He wanted to shoot extra Hobbit scenes next year that weren't originally included in the first two films.
"I'd like to shoot a bunch more material that we [couldn't] shoot. There's so much good stuff in the appendices that we haven't been able to squeeze into these movies."
Jackson announced he had finished shooting the films on July 6 with this Facebook post: "We made it. Shoot day 266 and the end of principal photography on The Hobbit."
He showed fans 12 minutes of material at Comic Con, including a chillingly comic exchange between Bilbo and Gollum, a tender moment between McKellen's Gandalf and Cate Blanchett, reprising her role as elf queen Galadriel, and the pivotal moment when Bilbo discovers the ring.
Jackson shot The Hobbit in 3D and at 48 frames a second, twice the speed that has been the standard since the 1920s. The higher frame rate allows for greater visual clarity, though it requires costly upgrades to digital projectors for cinemas showing films at that speed.