According to the Orange County Register, the full council will consider that request from Councilman Bruce Whitaker next Tuesday.
But the O.C. District Attorney's office told the newspaper the investigation into Thomas' death would be compromised, by releasing the video this early.
Two Fullerton police officers are facing charges in Thomas' death. Prosecutors say Thomas was beaten and suffocated.
Meanwhile, Fullerton's Police Chief Michael Sellers, who went on medical leave after the Thomas incident, is retiring.
Sellers went on leave in August, at the height of the public outcry over Thomas' death.
Sellers will be paid more than $127,000 in a workman's compensation settlement, as well as more than $22,000 in legal fees. His retirement will be effective Feb. 18.