A theory (MPL) to compute the NMR chemical shifts in condensed matter systems using periodic boundary conditions was presented by F. Mauri, B. Pfrommer, and S. G. Louie [Phys. Rev. Lett. 77, 5300 (1996)]. The MPL method has been implemented so far within a pseudopotential formulation in which the wave functions are expanded in plane waves. In this paper, we compare analytically the MPL approach within the density functional theory to existing methods for the calculation of the chemical shifts such as GIAO (gauge-including atomic orbitals), CSGT (continuous set of gauge transformations), and IGAIM (individual gauges for atoms in molecules). To this end we apply the MPL approach to molecules since the latter methods are conceived only for finite systems. We show theoretically the equivalence between a variant of the CSGT and the MPL method applied to finite systems. Moreover, we analyze numerically the efficiency of the different methods when atomic orbital basis sets are employed, by comparing the basis-set convergence properties. We find that the CSGT and IGAIM approaches have the same convergence properties as GIAO, whereas their computational time is significantly smaller. In the MPL method, the contribution of the valence electrons to the chemical shift converges rapidly with respect to the size of the basis set, whereas the convergence properties of the core contribution are poor. We improve the convergence by separating the core and the valence contributions in a gauge-invariant manner, by applying the MPL method only to the valence contribution, and by treating the core contribution with IGAIM. The performances of the resulting approach compare favorably with those of the other methods. Finally we find that the core contribution to the chemical shift is independent of the chemical environment, in contrast to what is sometimes found in the literature. In conclusion, our results indicate that, to compute the chemical shifts in both molecules and solids, using atomic orbital basis sets, one could use the MPL method to evaluate the valence contribution and add to it a rigid core contribution as obtained, for instance, from an atomic calculation.