TRACON Controller reported that the military trainer informed ATC an NMAC occurred with a VFR aircraft in the pattern. Upon further inspection; sufficient VFR to IFR separation was found and the VFR aircraft was instructed to maintain visual separation from the military trainer.

Narrative

I was conducting on-the-job training and my developmental and I just took the position. ZZZ Tower requested release for Aircraft X and the developmental released them. Aircraft X checked in with us runway heading off Runway XX climbing to 2000 ft. There was a VFR target in a right downwind for Runway XX that was going to pass about half a mile off the right side of Aircraft X. The developmental turned Aircraft X left to a 050 heading on initial contact; which turned the aircraft away from the VFR target and the two aircraft were no factor. The VFR target had no Mode C so we could not see what altitude they were at. After starting a turn to a 050 heading; Aircraft X reported that they responded to a traffic alert and climbed for evasive action but were finished and reported descending back down to 2000 ft.Aircraft X did not report an RA but about five minutes later asked us to mark the tapes for the NMAC they had on departure. I asked them if ZZZ Tower called the traffic to them; and they responded yes but that they did not have the aircraft in sight when told to contact Departure. Aircraft X said they received a traffic alert but did not receive an RA. I reported this information to the CIC (Controller in Charge) and they called ZZZ Tower for more info. ZZZ Tower did issue the traffic and told Aircraft X that the aircraft was on a right downwind at 1900 ft. ZZZ Tower switched Aircraft X to Departure without ensuring Aircraft X had the traffic in sight. The other aircraft told ZZZ they had Aircraft X in sight and reported level at 1200 ft. after Aircraft X was switched to Departure. Aircraft X reported a NMAC but the two aircraft were separated by 1/2 mile and 800 ft. within Class D airspace; as one was on downwind and one was on upwind. Aircraft X never saw the aircraft and based their decision on a traffic call by ZZZ Tower. However; the two aircraft were never in conflict with each other. By the time Aircraft X checked in with us; we did not know they were issued a traffic alert by ZZZ Tower. The trainee issued a left turn away from the traffic so in my opinion; a traffic call from us was not warranted as it was no factor. Had we known they were looking for traffic; we would have updated them that traffic was no factor. This event was caused by ZZZ Tower not updating Aircraft X on a previously issued traffic call. ZZZ Tower should have told Aircraft X the traffic was no factor or ensured they had the VFR target in sight before switching them to Departure.