Dave Diamond, a cable TV producer from Dayton, Ohio, wrote to Crist last month asking for the pardon. Diamond said the goal is to remember the Melbourne, Fla., native as an artist, not a rock 'n' roll bad boy with a rap sheet.
Crist is an alumnus of Florida State, which Morrison had attended.
"Well, given that fact, I'm certainly willing to review it," said Crist.
Morrison was charged days after a concert at Dinner Key Auditorium in Coconut Grove in 1969. He alleged exposed himself and simulated a sex act, which he denied doing.
He was acquitted on a felony charge for lewd and lascivious behavior, but was convicted of indecent exposure and profanity.
Many feel the Miami arrest put Morrison and the band in a downward spiral, which led to promoters canceling concerts and earned the band a stream of negative publicity.
Diamond and other fans say Morrison wouldn't have been charged if the same situation occurred today.
"We'd just like to 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," said Diamond.
The hard-living Morrison died of heart failure in a Paris bathtub in 1971, while his case was still on appeal. He was 27.
Morrison's father, retired U.S. Adm. George S. Morrison, 87, who lives in California, said he would support a pardon.
Crist, however, cannot pardon someone by himself. He needs two of the three other members of the Florida Cabinet, which acts as the state clemency board. In addition, there are no procedures for posthumous pardons.
In his letter to Crist, Diamond noted the former New York Gov. George Pataki pardoned the late comedian Lenny Bruce on an obscenity conviction.
"It's not about Jim Morrison's image as the Lizard King or The Doors music. It's about a citizen of Florida who was convicted in a case where the law was not applied," said Diamond, 34.