Police say Chavez has family in the West Covina area. The officer is said to be in the hospital and recovering from a chest wound that is considered non-life-threatening.
Chavez is described as a 47-year-old Hispanic, about 6 feet 1 inch tall, weighing about 225 pounds. He has dark hair, a mustache and a left eyebrow piercing.
The shooting occurred after officers stopped a man on a bicycle Friday night.
The suspect fired a shot at the officer, striking him in the chest, officials said. Officers returned fire and the suspect fled from the scene. West Covina, other police departments and dozens of SWAT officers spent more than 16 hours searching for the suspect, but have not located him.
Officials say they have a "strong indication" the suspect was struck in the exchange of gunfire. There was some blood from the scene that officials will test to determine if it belonged to the suspect or the officer.
The officer has been on the West Covina force for a year and a half.
Police detained and then released four people they thought may have had information about the shooting and Chavez's whereabouts.
Police asked anyone with information to call Los Angeles County Sheriff's Homicide at (323)890-5500 or call an anonymous tip to Crime Stoppers at (800)222-8477.