SR20
Flight Crew Multiengine
Time Pressure
Conflict NMAC
Air Traffic Control Provided Assistance; Flight Crew Requested ATC Assistance / Clarification; Flight Crew Returned To Clearance; Flight Crew Took Evasive Action
Human Factors
While flying on an IFR flight plan were being vectored for the RNAV 31 at ZZZ. After receiving and turning to a vector to intercept the approach course; upon leveling the aircraft we saw a Bonanza at our same altitude; within 50 meters of our nose crossing in front of us from left to right. The Bonanza was close enough I could see the pilot and clear aircraft detail. Immediately the student and I pitched and banked away from the aircraft to maneuver away. Right afterward; ATC asked us if that aircraft was at our same altitude. I then stated that he was and reported a 'near miss'. The rest of the flight resumed and we completed an approach at ZZZ. Enroute back to ZZZ1; ATC provided us with a phone number and instructed me to contact them after landing. I promptly called after landing and gave them my oral narrative of what happened.I think that perhaps ATC is being overworked. Throughout the flight we did receive many traffic advisories. However there was none for this event. Neither the student or I would have been able to see the Bonanza during the vector turn because we were busy setting up for the approach; and turning. To my knowledge the Bonanza was VFR and not talking on frequency.
SR-20 Instructor Pilot reported an NMAC during initial approach with an unreported aircraft.