Convicted of exposing himself during a Miami concert in 1969, Morrison was was sentenced to six months in jail. He died before his appeal was heard.
Dave Diamond of Dayton, Ohio, who asked Crist for the pardon, said the goal is to have Morrison remembered as an artist, not guy with a rap sheet.
"We'd just like to really see where we get to a point to let Morrison be judged because of the music and poetry and not be judged because of this Miami case," Diamond said.
Crist can't pardon someone by himself; two of the three members of the Florida Cabinet, which acts as the clemency board, must agree.
A spokeswoman for the state Parole Commission said no one is sure if a posthumous pardon has ever been considered before.
During the concert, Morrison rambled, swore and exhorted the audience to have sex. He also allegedly pulled down his pants and faked masturbation, which he denied. He was acquitted of a felony charge of lewd and lascivious behavior.
2014: The Year in Music [PHOTOS]