Commercial Fixed Wing
Flight Crew Multiengine; Flight Crew Instrument; Flight Crew Air Transport Pilot (ATP)
ATC Issue All Types; Conflict NMAC; Inflight Event / Encounter Weather / Turbulence
Air Traffic Control Provided Assistance; Flight Crew Took Evasive Action
Human Factors; Procedure; Weather
We were cleared for the visual approach to XXL into ZZZ. It was a very turbulent evening. I had briefed First Officer (FO) that my technique was to make my base turn overhead ZZZ1 at between 3;000 and 4;000 feet and then connect with the RNAV Visual to cross intersection ZZZZZ at 2;000 feet. This approach had worked well for me many times. As we were turning overhead ZZZ1; we heard a corporate jet tell TRACON that they had just departed ZZZ1. I believe it was a Cessna Citation. The controller hesitated and said 'standby for radar contact'. At the time; I was turning towards ZZZZZ at around 2;600 feet and descending. I was hand flying and concentrating ahead because of the turbulence; reported windshear and the fact that a previous Airbus had missed for windshear. The FO looked out of his window and said he saw the Citation. I then caught sight of the corporate jet off to our right and slightly below us. The FO said he thought they were turning towards us and closing in quickly. I pushed the nose down abruptly and at the same time received a TCAS Monitor Vertical Speed. I pushed down to approximately 1;900 feet and saw the Citation fly over us. The FO and I estimate we got to within 500 feet vertically. We remember seeing the TCAS show 600 feet separation. The controller quickly came onto the frequency and apologized; saying he thought the Citation was departing and flying to ZZZ VOR (a straight out departure) not towards us. The controller seemed confused. The FO told him that we had the aircraft in sight and he transferred us to Tower; apologizing again. Because of the extremely rough conditions; the passengers and flight attendants were all seated and buckled in and none noticed the extra push down to avoid the Citation Jet. There were no injuries reported in the cabin.
Air carrier Captain reported an airborne conflict on approach due to ATC communication misunderstanding with departure traffic.