Air Traffic TRACON Controller reported while sequencing aircraft to the airport they vectored an aircraft towards a VFR target that resulted in a NMAC. The pilot reported the NMAC and was vectored to the airport where they landed.

Narrative

I had just taken the position and Aircraft X was being vectored northbound on a downwind wanting to land R17R. There were two other jets [Aircraft Y] inbound that were much faster and had not been slowed so I decided that Aircraft X was going to be number three in the sequence but I didnt want to continue dragging him northbound. I told him to fly eastbound heading 080; short delay vector before I planned to turn him back to follow traffic. He was on his own altitude showing 075 and I noticed there was VFR traffic just north of FLY airport indicating around 077. When they were just over a mile apart I turned Aircraft X to a northbound 350 heading for that traffic. The VFR target also turned north unexpectedly and did appear to be closing so I called traffic but did not get any response from Aircraft X. The VFR target continued a turn to the south and I determined it was no longer a factor. When I called the Jet traffic [Aircraft Y] passing above Aircraft X as his traffic to follow his response was that he was recovering from a near mid air and wanted a phone number to call the TRACON and report the incident. I continued to vector him in to R17R and gave him the TRACON phone number. He landed without further incident. Suggestion: I did see the VFR target Aircraft X was in conflict with over FLY and thought I had given the northbound turn in time to deconflict. I did notice their target continued to follow and converge. I called traffic but got no response from Aircraft X. In hindsight I think I should have called a traffic alert instead of just traffic. Because I received no response after my initial traffic call Im not really sure if I got his attention or not. A traffic alert call might have gotten his attention quicker. The VFR target did continue southbound away from Aircraft X and I was back to thinking about fitting him into the sequence.