For drivers, the limit is 0.08 percent in most states.
The controller is no longer handling air traffic, the agency said. His name wasn't released.
It wasn't immediately clear what discipline, if any, he would face if the allegations are upheld.
The story was first reported by KMGH-TV.
The National Air Traffic Controllers Association, which represents controllers, called the claims "deeply troubling."
"We take our responsibility of ensuring aviation safety very seriously. That includes acting professionally in all that we do," association president Paul Rinaldi said in a written statement.
The reported violation occurred at the Denver Air Route Traffic Control Center, which directs airplane travel between airports but not in the immediate vicinity of airports. The center is located in Longmont, about 25 miles north of Denver.
It came just days after the FAA announced new steps to help prevent controllers from falling asleep on the job. Since April, the FAA has disclosed seven instances of controllers sleeping while on duty.
The new policy allows controllers to use sick or annual leave time if they are too tired to work. They will also be allowed to listen to the radio and read to help stay alert during overnight shifts when traffic is light.
The new policy doesn't allow controllers to nap while on break or to schedule naps during overnight shifts, even though sleep scientists say that's the most effective way to refresh tired workers.
Currently, controllers caught napping, even when on break, can be fired.