The occurrence of spherical shell closures in the superheavy nuclei region is explored in the framework of the relativistic Hartree-Fock-Bogoliubov theory (RHFB), and also with the simpler version of relativistic mean field with quasiparticles, in other words the relativistic Hartree-Bogoliubov (RHB) approximation. Both theoretical frameworks are often used for nuclei in the known regions of the nuclear chart. Shell closures characterized in terms of two-nucleon gaps indicate the magic regions. The results depend slightly on the effective Lagrangians used, but the magic numbers beyond 208Pb are generally predicted to be Z = 120, 138 for protons and N = 172, 184, 228, 258 for neutrons. Shell effects are sensitive to various terms of the mean-field, such as the spin-orbit coupling, the scalar and effective masses. This explains the (relatively small) variations in the predictions, depending on the effective Lagrangians employed. The most complete model (RHFB) favors the nuclide 304 120 as the next spherical doubly-magic nucleus beyond 208Pb.