Flight Instructor on a training flight with student were cleared for takeoff by the tower when another aircraft was on short final. This resulted in a near miss between these two aircraft.

Narrative

I was holding short of Runway XXR at the intersection of ZZZ in a Cessna 172; on a sunny afternoon in good weather and fair visibility. The Tower controller was working two frequencies; and we called holding short and requested a departure. The controller immediately cleared us for takeoff on [Runway] XXR; which my Private Pilot student read back correctly. We started moving across the hold short bars when another aircraft in the pattern; a Cessna 152; called 'Aircraft Y on short final.' I slammed on the brakes; and a few seconds later the Cessna crossed over the numbers directly in front of me at approximately 40 feet in front of my nose and 50 ft. AGL. At this point we were inside the Runway Safety Area (RSA) and about halfway past the hold short markings about to enter the landing surface. If I had not slammed on the brakes; the short final traffic had not called on frequency; and/or there had not been a displaced threshold a collision may have occurred. The geometry was such that I was not able to see the aircraft on short final before moving into the RSA. This was the nearest miss of my piloting career and was a result of the Tower controller clearing me for takeoff with another aircraft on short final.