Yellow Cab made the request at the beginning of 2013, according to Joan Adcock, a Little Rock city director.
Adcock said she then requested the city look into the change, which would reflect similar policies in other cities.
The proposed ordinance would allow people convicted of non-violent felonies to get a taxi cab driver permit if the charge is more than five years old.
The applicant would not be considered if they've been convicted of murder, kidnapping, battery in the first or second degree, sexual offense or robbery.
It adds that additional criminal offenses can disqualify the applicants if they were also convicted of a misdemeanor in the past two years.
Adcock said she has not yet heard any opposition, adding it would give some folks a second chance.
But there is some concern the additional drivers could increase the already-competitive market for cab drivers.
"We already have more cab drivers than we need," one taxi cab driver, who did not want to be identified, told Channel 7. "A lot of drivers end up dropping their cabs and doing something else for the summer."
The ordinance will be put on the agenda Tuesday for next week's meeting.
It could be voted on as early as next week.