A reamer has a head with a recessed bearing plane which absorbs the pressure of a clamping member which presses against one side of a cutter knife. A counter member is positioned with one side displacable on the recessed bearing plane in the direction of increased cutting diameter. The cutter knife is held between the clamping member and the counter member. The counter member has a region with a straight holding portion which supports the cutter knife. The clamping member and counter member have congruent oblong holes which extend in the direction of increased cutting diameter. These holes are traversed by a clamping screw which is screwed into a threaded hole in the head. When the screw is tightened, the three piece unit of clamping member, cutter knife and counter member are tightened in the reamer head. A pair of set screws pass through holes in the head to move contact pieces against a thrust bar which extends from the straight holding portion of the counter member, and hence move the three piece unit in the direction of increased cutting diameter.