Flight Instructor with student practicing maneuvers reported an NMAC with an aircraft on a VOR approach with the VOR out of service. Flight Instructor took evasive action to avoid a collision.

Narrative

I was flying VFR with a student pilot; at 5500 MSL approx. 14 NM NW of ZZZ (open area). I heard Aircraft Y on Tower frequency for ZZZ - as they informed Tower they were on the VOR XX Approach (which Tower questioned; as the VOR is NOTAM'd out of service until Day 0). The pilots of Aircraft Y definitely had a hard time trying to cover their mistake on the radio.Thinking nothing of it; my student and I were practicing Unusual Attitudes (her under the hood); and when I glanced at my ADS-B - I saw Aircraft Y heading our general direction. I began to turn away from them; and they appeared to speed up; and continued directly for us; at our altitude. They were going westbound.I spotted them in the air; attempted to turn away from their line of travel; and ultimately; our Garmin GTN 650 gave a traffic alert; prompting an emergency/evasive descent to prevent a midair. I don't think they saw us once - as they continued on as if we weren't there. (Which fits; as they practiced a VOR approach without a VOR...)