GA flight instructor with student reported an NMAC in OKC Class E airspace requiring evasive action to avoid a possible collision.

Narrative

IFR training flight out of RWY 17L PWA received vectors for climb out from TRACON. Level at 4000 feet msl on a heading of 320 we were given a traffic advisory from ATC there was an aircraft was ahead to our 1 o'clock at 3700 feet and 1 mile. I advised ATC we did not have traffic in sight. Several seconds later we received a 'traffic' warning from the G1000 Traffic Information Service indicating traffic was ahead and 500 feet below us. About 3 seconds later when the traffic continued to move closer and when the altitude separation indicated 300 feet I elected to make an immediate left 360 degree turn. Never made visual contact with traffic. ATC queried our turn; we advised them we were making a maneuver for traffic avoidance due to an in-aircraft advisory message. We were told to advise when we could continue with headings. On completion of the 360 degree turn we resumed previous assigned 320 heading and advised ATC we were on course.As the instructor onboard I was concentrating on making sure the student was in compliance with ATC clearances; I could have done a better job of scanning for traffic in VMC and dividing my attention between inside and outside of the aircraft. With the TIS information we had displayed on our MFD I was relatively certain of the other aircraft's position and direction of flight. I believe that a turn away from the aircraft's indicated position as well as the information we received from ATC was the correct course of action as we noted the aircraft getting closer without any further advisories from ATC but had insufficient time to ask ATC for new headings before the in-aircraft advisory was received.