We carry out an ab initio study of the structural, electronic, and magnetic properties of zigzag graphene nanoribbons on Cu(111), Ag(111), and Au(111). Both, H-free and H-terminated nanoribbons are considered revealing that the nanoribbons invariably possess edge states when deposited on these surfaces. In spite of this, they do not exhibit a significant magnetization at the edge, with the exception of H-terminated nanoribbons on Au(111), whose zero-temperature magnetic properties are comparable to those of free-standing nanoribbons. These results are explained by the different hybridization between the graphene 2p orbitals and those of the substrates and, for some models, also by the charge transfer between the surface and the nanoribbon. Interestingly, H-free nanoribbons on Au(111) and Ag(111) exhibit two main peaks in the local density of states around the Fermi energy, which originate from different states and, thus, do not indicate edge magnetism.