The crash happened around 1 a.m. Monday in the 2400 block of Fannin Street.
Houston police say there was an inmate in the back of the police vehicle when the driver ran a stop sign and T-boned them.
The inmate suffered a head injury and was taken to Memorial Hermann Hospital, HPD said.
Officers are working to determine if the driver was intoxicated at the time of the crash.
"If he's intoxicated, it would be a DWI," Sgt. J. Uribe said about possible charges. "If he's not intoxicated, he'll be written up for running a stop sign."
The HPD officers involved were not hurt, officials say.
The inmate is expected to make a full recovery.