The coupling between two atomically sharp nanocontacts provides tunable access to a fundamental underlying interaction: the formation of the bond between two atoms as they are brought into contact. Here we report a detailed experimental and theoretical analysis of the relation between the chemical force and the tunneling current during bond formation in atom-scale metallic junctions and their dependence on distance, junction structure, and material. We found that the short-range force as well as the conductance in two prototypical metal junctions depend exponentially on the distance and that they have essentially the same exponents. In the transition regime between tunneling and point contact, large short-range forces generate structural relaxations which are concomitant with modifications of the surface electronic structure and the collapse of the tunneling barrier.