Earlier this year, the Presidio Trust rejected Lucas' proposal to open the museum on park land near the Golden Gate Bridge. He's still considering the Presidio Trust's offer of another location near the home of Lucasfilm's special-effects, game unit and corporate offices.
The museum will feature the "Star Wars" creator's lifetime art collection and movie memorabilia.
The city of Chicago is launching an aggressive campaign to bring the museum there, and is considered a front runner for the display of Lucas' collection in part because he is married to Chicago businesswoman Mellody Hobson.
In a statement, Mayor Lee said he is well aware that San Francisco is in competition with another city, "I will not let go easily of such a significant private investment by one of the world's most prominent filmmakers," he said.
Mayor Lee says he has given his staff until the end of this month to come up with possible locations for the museum.
(The Associated Press contributed to this report)