Pilot reported a near midair collision on final to fnl airport.

Narrative:

My student and I had been cleared by denver approach for a visual approach into fnl. Approach instructed us to switch to tower and we did (fnl is towered east airspace). My student told tower we were from the south. Tower asked us if we were setting up for a right downwind for runway 15. We confirmed we were. Tower instructed us to report 2 miles south of the airport. They asked us if we had traffic in sight and we confirmed that we did. Tower told us we were number 3. I saw on the traffic map that there was one aircraft on final and another on downwind. I assumed these two planes were the aircraft that tower was referring too. We were on midfield right downwind about 2 miles south and reported to tower where we were. Tower asked us if we had upwind traffic insight and to follow them. I clarified with tower that we were already on downwind ahead of that aircraft. Tower cleared us to land runway 15; number 2. I verified on the traffic map that there was one aircraft in front of us and visually as well. I saw that there were at least two aircraft behind us in the pattern. I told my student to wait till the aircraft was past our abeam point; as we were on downwind and the aircraft ahead of us was on final; to provide more spacing between us. Once the aircraft was past our abeam point my student turned right base. My student rolled wings level and we both checked final for aircraft; neither of us saw any. My student turned final. We were about 500-600 feet AGL when aircraft Y appeared from under our left wing in front of us. My student configured the plane for a go-around as I told him to immediately turn to the north. I called tower and told them that an aircraft had cut us off and that we were departing to the north. Tower came back and told us to do a 270 turn. We complied. As we were turning I remembered and saw the two aircraft on downwind. We called tower to confirm that we were number 1 and cleared to land runway 15. Tower confirmed. We landed with no further issues. Tower didn't say anything to aircraft Y or us. I never heard aircraft Y make any calls to tower or tower to them. Aircraft Y never appeared on our traffic map and we never got a traffic alert. I believe there was a miscommunication between tower and aircraft Y and aircraft Y thought they were cleared to land. To prevent this occurrence from happening again I believe tower needs radar or some other form of technology that shows their airspace and the aircraft in it.

Original NASA ASRS Text

Title: Pilot reported a NMAC on final to FNL airport.

Narrative: My student and I had been cleared by Denver Approach for a visual approach into FNL. Approach instructed us to switch to Tower and we did (FNL is towered E airspace). My student told Tower we were from the south. Tower asked us if we were setting up for a right downwind for Runway 15. We confirmed we were. Tower instructed us to report 2 miles south of the airport. They asked us if we had traffic in sight and we confirmed that we did. Tower told us we were number 3. I saw on the traffic map that there was one aircraft on final and another on downwind. I assumed these two planes were the aircraft that Tower was referring too. We were on midfield right downwind about 2 miles south and reported to Tower where we were. Tower asked us if we had upwind traffic insight and to follow them. I clarified with Tower that we were already on downwind ahead of that aircraft. Tower cleared us to land runway 15; number 2. I verified on the traffic map that there was one aircraft in front of us and visually as well. I saw that there were at least two aircraft behind us in the pattern. I told my student to wait till the aircraft was past our abeam point; as we were on downwind and the aircraft ahead of us was on final; to provide more spacing between us. Once the aircraft was past our abeam point my student turned right base. My student rolled wings level and we both checked final for aircraft; neither of us saw any. My student turned final. We were about 500-600 feet AGL when Aircraft Y appeared from under our left wing in front of us. My student configured the plane for a go-around as I told him to immediately turn to the North. I called Tower and told them that an aircraft had cut us off and that we were departing to the North. Tower came back and told us to do a 270 turn. We complied. As we were turning I remembered and saw the two aircraft on downwind. We called Tower to confirm that we were number 1 and cleared to land Runway 15. Tower confirmed. We landed with no further issues. Tower didn't say anything to Aircraft Y or us. I never heard Aircraft Y make any calls to Tower or Tower to them. Aircraft Y never appeared on our traffic map and we never got a traffic alert. I believe there was a miscommunication between Tower and Aircraft Y and Aircraft Y thought they were cleared to land. To prevent this occurrence from happening again I believe Tower needs radar or some other form of technology that shows their airspace and the aircraft in it.

Data retrieved from NASA's ASRS site and automatically converted to unabbreviated mixed upper/lowercase text. This report is for informational purposes with no guarantee of accuracy. See NASA's ASRS site for official report.

Attributes
ACN	1746619
Time
Date	202006
Local Time Of Day	0601-1200
Place
Locale Reference	FNL.Tower
State Reference	CO
Environment
Flight Conditions	VMC
Light	Daylight
Aircraft 1
Make Model Name	Small Aircraft
Operating Under FAR Part	Part 91
Flight Phase	Final Approach
Route In Use	Visual Approach
Flight Plan	IFR
Aircraft 2
Make Model Name	Any Unknown or Unlisted Aircraft Manufacturer
Flight Phase	Landing
Person 1
Function	Instructor Pilot Not Flying
Qualification	Flight Crew Commercial Flight Crew Flight Instructor Flight Crew Instrument
Experience	Flight Crew Last 90 Days 73 Flight Crew Total 315 Flight Crew Type 315
Events
Anomaly	Conflict NMAC
Miss Distance	Horizontal 130 Vertical 70