The reaction of diazo compounds with transition-metal carbenes is an efficient way to achieve the functionalization of chemical bonds in organic molecules, especially for the C-H and O-H bonds. However, the selective mechanisms of C-H and O-H bond insertions by various metal carbenes such as Rh and Cu complexes are not quite clear. In this work, we performed a comprehensively theoretical investigation of the phenol C-H and O-H bonds inserted by Rh and Cu carbenes by using DFT calculations. The calculated results reveal that the nucleophilic additions of phenols to the Rh and Cu carbenes in the C-H bond insertions are the rate-determining steps of whole reactions, which are higher than the barriers in the O-H insertions. In the process of intramolecular [1,3]-H transfer, the Rh and Cu ligands in their carbenes tend to dissociate into solution rather than the intramolecular migration due to their weak metal,carbon bonds. A deeply theoretical analysis of the electronic structures of Rh, Cu, and Au carbenes as well as their complexes elucidated their differences in the chemoselectivity of C-H and O-H insertion products, which agrees with the experimental observations well.