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 $\chi^*$ from the Shubnikov-de Haas oscillations in crossed magnetic fields and (ii) thermodynamic susceptibility $\chi_{\rm T}$ sensitive to all the electrons in the layer. Both $\chi^*$ and $\chi_{\rm T}$ 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 $n_c$. Moreover, the value of $\chi_{\rm T}$, which can be measured across the transition down to very low densities deep in the insulating phase, increases with density at $n<n_c$, as expected. In the absence of magnetic field, we found the temperature dependence of $\chi^*$ 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 $\chi_{\rm T}$ to localized spin droplets. In strong enough in-plane magnetic field, we found the temperature dependence of $\chi^*$ to be stronger than that expected for the Fermi liquid interaction corrections.