DA40 pilot with instructor reported an NMAC while practicing an ILS approach with an aircraft with no Mode C; nor ADS-B. Pilot took evasive action to avoid a possible collision.

Narrative

I was returning to ZZZ1 from the practice area as part of a commercial training flight. I was the student and pilot flying; my CFI was pilot monitoring. Passing just to the NNE of the ZZZ airport; we were receiving flight following from Approach; on an ATC-assigned heading SSE; and adhering to a VFR altitude restriction at 3;000 feet. Our intention was to continue receiving ATC vectors to set up for a practice ILS approach to Runway XX at ZZZ1. During my periodic scan; I suddenly noticed another aircraft just ahead and to the left of us; passing roughly east to west and converging. It was roughly the relative size of my fist; so I estimate an approximate 300 foot distance. I immediately disconnected the autopilot and took evasive action by entering a hard left climbing turn. We executed a 360 turn and rejoined our approximate prior heading. Pilot monitoring called ATC to advise of the other aircraft and our intentions since we were forced to deviate from ATC-provided instructions. ATC initially advised that they didn't see another target in our vicinity but then followed up to say a primary target had shown up; but no altitude. The aircraft I was flying is equipped with ADS-B in and at no point did we receive traffic information for this target either. While not regulatory for the other aircraft to have Mode C in this airspace; had the other aircraft been equipped and/or transmitting Mode C or ADS-B Out; it would have greatly increased overall situational awareness for both other AC and ATC; especially on a day where haze can degrade see-and-avoid.