The "Hannibal" showrunner will write and executive produce a new version of the series, reported The Wrap.
Spielberg, 68, who also served as the executive producer of the original series, will not be involved in the remake.
The series, which ran for two seasons in 1984-85, featured new sci-fi and supernatural stories each episode.
Fuller, 46, is currently busy working on adaptation of Neil Gaiman's "American Gods.