A Tower Local Controller reported an aircraft on straight in final approach reported a NMAC when an aircraft on downwind turned into them.

Narrative

I was working the Local 1 position for Runway XXR. I had several aircraft inbound; and in the pattern. Aircraft X had turned approximately a 3 mile final for Runway XXR; and Aircraft Y had been following them for several miles on the right downwind; and had reported them in sight. I heard Aircraft X say there was an aircraft close off their right side; so I immediately told Aircraft Y; to turn right and fly south. After Aircraft Y acknowledged; Aircraft X said; 'that was really close; less than 1/4 NM at same altitude.' Aircraft X had also climbed to 6500 ft. to avoid Aircraft Y. I re-sequenced them both back into the pattern; and both landed safely. I believe that the pilot of Aircraft Y lost sight of Aircraft X in the setting sun; and assumed when to turn rather than ask for direction. This was a fluke caused by the sun blinding a pilot; and an unwillingness by the pilot to admit he had lost the traffic. If the pilot would ask for a sequence; or even just say they lost their traffic; I could have re-sequenced them with no issues.