The proximity-induced effects of the nearly commensurate lattice structure of a graphene layer AC stacked on Ni(111) and Co(0001) substrates are addressed. To this end, a minimal tight-binding Hamiltonian is constructed within the Slater-Koster method. We consider the hybridizations of the magnetic 3d orbitals of Ni (Co) atoms with the pz orbitals of graphene, in addition to the atomic spin-orbit coupling and the magnetization of the Ni (Co) atoms. A derivation of the low-energy perturbed π bands in the vicinity of the Dirac points enables us to get further insight into the physical nature of the induced effective couplings to the graphene layer. It is shown that a magneto-spin-orbit type effect may emerge through two competing mechanisms simultaneously present, namely, the proximity-induced exchange and Rashba spin-orbit interaction. Such effects result in giant exchange splittings and robust Rashba spin-orbit coupling transferred to the graphene layer in agreement with recent density functional theory calculations and experimental observations. We further analyze the physical conditions for the appearance of intact Dirac cones in the minority spin bands as observed with recent photoemission measurements with spin resolution.