Adsorption of NH3 and BBr3 (and their decomposed fragments) onto B and N sites of c-BN(111) and h-BN(100), has been theoretically investigated using DFT calculations under periodic boundary conditions. The purpose was to study the adsorption processes of the precursors during gentle deposition of BN thin films in order to find prerequisites for ideal growth of c-BN, and to simultaneously discriminate h-BN growth. The present results show that adsorption of NH3, NH2, NH, N, BBr3, BBr2, BBr, and B (with three exceptions) is exothermic to either the B or N sites of c-BN(111) and h-BN(100), respectively. The exceptions include adsorption of NH3 to the N site of either the cubic surface or the hexagonal edge, as well as adsorption of BBr3 to the B-rich surface of c-BN. In opposite to all other combinations studied, adsorption of NH3 and BBr3 was found to preferably lead to an ideal growth of c-BN.