Center Controller reported an NMAC when a VFR military trainer was at an IFR altitude and came close to an IFR aircraft.

Narrative

Aircraft X was eastbound from ZZZ Approach level at 070. There was an aircraft squawking XYXY that was in a CA with Aircraft Y at 069 flying northeast and maneuvering. There was an aircraft nearby squawking a similar code (possibly XXXY) that was maneuvering as well. I issued the traffic to Aircraft X which was approximately 6 miles away and told the pilot that the aircraft is currently flying northeast. I also told him that I am not sure what instruction to give him to help him in the situation since the altitude is unverified and if I turned him right (which would put the aircraft behind the current flight path of the XYXY code) and then the other aircraft turned to the left; it would turn him right into the other aircraft. I then cleared the aircraft to maneuver vertically and horizontally as necessary to ensure he kept clear and instructed to remain at or above 032.Within the next couple of minutes; I updated Aircraft X on the traffic and that the targets are likely to merge and we needed to do something. Another aircraft tried checking on so I said to maintain radio silence and returned to Aircraft X to issue a traffic alert. I gave him the position of the aircraft in the traffic alert but no control instruction because I wasn't sure what the other aircraft was going to do since it was maneuvering laterally and vertically. At some point; the pilot informed me that he was climbing and I observed 074 in the data block. The targets merged with approximately 500 feet of separation. Once the aircraft was behind Aircraft X I informed him of the other aircraft's position and gave him discretion to return to 070 when he felt it was safe.I then called ZZZ Approach and asked if they were talking to an aircraft east of the field squawking XYXY to which they said that they weren't. Shortly thereafter; Approach used the landline to call and say that they were and I told my D Side that I wanted to talk to that aircraft and put them on me. A little bit of time went passed and they still hadn't put the aircraft on me so my D Side asked once again. Shortly thereafter Aircraft Y came on my frequency and I asked him if he was in the aircraft squawking XYXY. The rest of this portion isn't exact wording or everything that was said on frequency but I will give the gist of it. He told me no that it was his wingman.I explained to the pilot that he had just merged with Aircraft X at the IFR altitude of 070 and that I understand they are training but if they can accomplish their mission at VFR altitudes that would be better. He said that he knew that aircraft was there and they try and frequently change altitudes. I said that if this situation happens again and he knows he is going to get that close to another aircraft while he is talking to ATC; that he should tell the controller so we know what is happening. I explained that I was unable to tell the other aircraft how to avoid the collision because I didn't know what Aircraft Y was doing. Aircraft Y also didn't know that he was in my airspace even though he was talking to Approach and that he was ask them to transfer their flight to us next time.Due to the very pleasant and professional conversation that I was able to have with Aircraft Y on frequency after the occurrence; I do believe that he will take the steps that I mentioned of telling whatever controller he's talking to that he plans on coming extremely close to another aircraft so that we can safely resolve the situation. Due to the difference in air frame; Aircraft X had no chance of getting away from the situation if Aircraft Y hadn't seen Aircraft X for himself. If there is formation work occurring outside of ZZZ Approaches airspace on VFR aircraft in a normal IFR corridor; Center needs to be in communication with that formation. While this situation ended well; it did not need to develop into a traffic alert situation and could have easy been mitigated if any of the aforementioned suggestions would have been done.