A TRACON Controller reported their Manager questioned them about a NMAC between an IFR aircraft they had handed off to the Tower and a VFR aircraft in the Tower's Class D airspace.

Narrative

I was working sector X; working departures of ZZZ and feeding arrivals to the ZZZ final Controller. I also work satellite airports southwest of ZZZ. I had just relieved the previous Controller from the position; which during the relief briefing; told me that he had cleared Aircraft X; for the RNAV XX approach at ZZZ1. When Aircraft X turned final on the approach; I switched him to ZZZ1 Tower. I observed a VFR at 1;900 ft approximately 6 miles southeast of Aircraft X heading directly towards ZZZ1. This is a common occurrence; because ZZZ1 Tower is a busy VFR Tower. I switched Aircraft X early enough so that ZZZ1 would be able to set their sequence. Both the VFR and IFR; Aircraft X; entered the class D at the same time. The VFR from the southwest and the IFR from the west. Having worked at a VFR Tower; this didn't alarm me or cause concern. Later; the ZZZ1 Tower Controller asked if I ever talked to Aircraft Y; which I hadn't. On Day 0; I was notified that this incident was being logged as a significant event and that Aircraft X had to conduct [priority handling] left 360 to avoid the VFR. I was interviewed by my operations manager. The pilot of Aircraft X stated that in his opinion; the VFR was in the class D airspace. Upon listening to the tapes; you do not hear a collision alert go off in the background while I'm talking to other planes. When I saw the left 360; I assumed that ZZZ1 was sequencing the VFR and IFR aircraft for ZZZ1. Suggestion: I don't know why TRACON Controllers are being interviewed for a significant event that happened in another facilities airspace; involving an aircraft that violated class D airspace. ZZZ1 Tower doesn't have a radar display at all. They only have the windows to find traffic. It was brought up before at ZZZ [TRACON]; how ZZZ1 not having a D-brite; or radar display; could lead up to a significant event. I think if a contract Tower can afford to open a Tower; they should afford to have a radar display. If not; it was safer having aircraft on UNICOM or with the radar.