This paper is part of a larger project in which we systematically study the chemical abundances of Galactic and extragalactic post-asymptotic giant branch (post-AGB) stars. The goal at large is to provide improved observational constraints to the models of the complex interplay between the AGB s-process nucleosynthesis and the associated mixing processes. Aims. Lead (Pb) is the final product of the s-process nucleosynthesis and is predicted to have large overabundances with respect to other s-process elements in AGB stars of low metallicities. However, Pb abundance studies of s-process enriched post-AGB stars in the Magellanic Clouds show a discrepancy between observed and predicted Pb abundances. The determined upper limits based on spectral studies are much lower than what is predicted. In this paper, we focus specifically on the Pb abundance of 14 Galactic s-process enhanced post-AGB stars to check whether the same discrepancy is present in the Galaxy as well. Among these 14 objects, two were not yet subject to a detailed abundance study in the literature. We apply the same method to obtain accurate abundances for the 12 others. Our homogeneous abundance results provide the input of detailed spectral synthesis computations in the spectral regions where Pb lines are located. We used high-resolution UVES and HERMES spectra for detailed spectral abundance studies of our sample of Galactic post-AGB stars. None of the sample stars display clear Pb lines, and we only deduced upper limits of the Pb abundance by using spectrum synthesis in the spectral ranges of the strongest Pb lines.