Bounty hunter Boba Fett became a cult favourite of Star Wars fans after his first appearance in The Empire Strikes Back, before his death in Return of the Jedi.
Johnston was a designer and visual effects art director of the first three Star Wars films and also designed Fett's famous outfit.
The Captain America: The First Avenger director told Screenrant that he has asked George Lucas about a possible spinoff, and that he would like to direct.
He said: "I'm trying to get George to make a feature based on Boba Fett. I would like to direct it. It would be a lot of fun."
Johnston, who directed Jurassic Park III, added that he has hopes of a fourth movie in the dinosaur franchise.
"We are in discussions. The most important thing I can tell you is that it starts a new trilogy that goes in a different direction. It starts with the history of the first trilogy but spins off in a completely different direction that's very exciting and different from anything we've seen. That's all I can tell you."
The director recently said that he would be interested in making a sequel to Captain America.
Watch Robot Chicken's spoof of Boba Fett in Return of the Jedi below: