The museum building has been designed by Chinese architect Ma Yansong, the LA Times reports, and has two potential locations: Treasure Island in San Francisco Bay and Exposition Park in Los Angeles.
The San Francisco Chronicle has reported that local officials are receptive to the Treasure Island site.
"I'm not an architectural expert, but I do like the cloud concept and I do think it's a spectacular site," Jane Kim, who represents Treasure Island on the Board of Supervisors, told the Chronicle.
The new building, which looks like either a spaceship or a cloud, will include theaters, a 4,200-square-foot drop-in library, classrooms and several restaurants.
The museum is designed to exhibit works from Lucas himself -- who got the whole "Star Wars" juggernaut started -- as well as from sources as diverse as ancient Greece, Norman Rockwell and Harry Potter.
The LA Times says Lucas is expected to choose a location for the museum early in 2017.