Spielberg directed Dakota in 2005's alien invasion epic War of the Worlds and serves as a producer on Elle's new movie with JJ Abrams, Super 8.
Fanning told Digital Spy: "I'd love to be directed by Steven because he's just a producer on this, that would be cool to experience that. My sister was in War of the Worlds and he directed that. That would be fun for me to do something like that as well."
The 13-year-old added that Spielberg was often present on set to act as a mentor to Abrams.
"JJ looked up to him so much. Steven was really involved and JJ, you always see him thinking about the details," she said.
"When you see the bedroom of the characters in the movie, you see all the posters on their wall and that's such a unique touch to things. I think JJ knew how to do that because he was actually one of those kids."
Super 8 opens in UK cinemas this Friday.
More Super 8 coverage:
> 'Super 8' review
> Elle Fanning 'Super 8' interview: "I let my inner zombie out"
> 'Super 8': Ten teasers about the sci-fi smash
> Watch a 6-minute 'Super 8' preview
Watch our full Elle Fanning video interview below: