We report temperature and density dependences of the spin susceptibility of strongly interacting electrons in Si inversion layers. We measured (i) the itinerant electron susceptibility χ∗ from the Shubnikov-de Haas oscillations in crossed magnetic fields and (ii) thermodynamic susceptibility χTsensitive to all the electrons in the layer. Both χ∗ and χT are strongly enhanced with lowering the electron density in the metallic phase. However, there is no sign of divergency of either quantity at the density of the metal-insulator transition nc. Moreover, the value of χT, which can be measured across the transition down to very low densities deep in the insulating phase, increases with density at n<nc, as expected. In the absence of magnetic field, we found the temperature dependence of χ∗ to be consistent with Fermi-liquid-based predictions, and to be much weaker than the power law predicted by non-Fermi-liquid models. We attribute a much stronger temperature dependence of χT to localized spin droplets. In strong enough in-plane magnetic field, we found the temperature dependence of χ∗to be stronger than that expected for the Fermi liquid interaction corrections.