Falcon 900
Flight Crew Air Transport Pilot (ATP); Flight Crew Multiengine; Flight Crew Instrument
Communication Breakdown; Confusion
ATC Issue All Types; Deviation - Altitude Overshoot; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter CFTT / CFIT; Inflight Event / Encounter Weather / Turbulence
General None Reported / Taken
Human Factors
While on approach to ZZZ we were cleared to descend to 3;000 feet at which time we had the runway in sight. Upon reaching 3;000 feet the controller gave us a low altitude alert and wanted to verify we were level at 3;600.  The PNF (Pilot Not Flying) responded that we were level at 3;000 and had the airport in sight. The controller advised he had only cleared us to descend to 3;600 not 3;000. We then called the airport in sight and he cleared us for the visual approach and we initiated a successful landing. After landing the Tower Manager contacted the flight crew via telephone at the FBO and advised there was a possible pilot deviation.
We were in contact with ZZZ Approach. We were to expect radar vectors for the ILS XX. During the descent there were thunderstorms in the area. The approach controller instructed to descend to 3;000. As we leveled at 3;000 the controller ask to confirm our altitude. I informed the controller we were level at 3;000. He informed us we should have been at 3;600. We informed him we thought he had said 3;000. By this time we were clear of clouds and has the airport in sight. We climbed to 3;600 and received radar vectors as we were cleared for the approach.
Flight Crew flying F-900 aircraft reported altitude deviation on approach.