This paper presents the possibility of forming a pressed joint between two 1.5 mm thick sheets made of HX340 steel. The joint was formed using an additional deformable steel rivet with a hardness of 400HV1. Different distances of rivet pressing below the initial surface of the upper sheet were chosen: 0, 0.15, 0.30, 0.45 and 0.60 (in mm). For the mentioned rivet-pressing distances, forming tests were performed using an “SKB” die with three die depths: 1.45, 1.60 and 1.75 (in mm). Observations and measurements of the characteristic geometric dimensions of the interlock were made. Furthermore, the strength of the clinch-riveted joints was analysed. Changing the depth of the die by 20%, i.e., from hd = 1.45 mm to hd = 1.75 mm for the same rivet press-in depth (of = 0%), resulted in a 12% reduction in the forming-force value. The maximum load capacity of the joint in the shear test decreased by 3.5% for the mentioned joint variant. The greater the rivet press-in depth used in the joint (of), the greater the value of the interlock size (tu). Increasing the value of rivet-pressing depth below the initial surface of the upper sheet from 0 mm to 0.6 mm resulted in an increase in the size of the interlock (tu) by 136% (for hd = 1.45 mm), by 128% (for hd = 1.60 mm) and by 85% (for hd = 1.75 mm).