I do avionics maintenance for a corporate flight department. One of our pilots had this same issue with a Bose ANR headset. In testing, I found that the noise would vary in intensity with head position, getting especially loud when the headset was brought in close proximity to a circuit breaker panel on the sidewall next to the pilot seat.
It turned out that the shield had broken inside the cable going to one of the two earpieces, causing the unit to pickup radiated noise from the aircraft wiring. The problem was actually worse when the headset's noise-canceling feature was turned on - probably because the induced noise was being amplified by the ANR circuitry.
I'd suspect a ground/shielding problem. Either in the headset itself, or in an associated adapter.