Aims: The aim of this work is to search for Li-rich giants in a sample of clusters where planets have been searched, thus we can study the planet engulfment scenario to explain Li replenishment using a proper comparison sample of stars without detected giant planets. Methods: We derived Li abundances for a sample of 67 red giant stars in 12 different open clusters using standard spectral synthesis techniques and high-resolution spectra (from HARPS and UVES). We also determined masses, ages, and radius from PARSEC stellar isochrones to constrain the evolutionary stage of these stars. Results: We found three stars in different clusters with clearly enhanced Li abundances compared to other stars within the cluster. Interestingly, the only two stars with a detected substellar companion in our sample belong to that group. One of the planet hosts, NGC 2423 No. 3, might lie close to the luminosity bump on the HR diagram, a phase where Li production by the Cameron-Fowler process is supported by extra-mixing to bring fresh Li up to the surface. On the other hand, NGC 4349 No. 127 is a more massive and more evolved giant that does not seem to be in the evolutionary phase where other Li-rich stars are found. We discuss the possibility that the Li enhancement of this star is triggered by the engulfment of a planet, considering that close-in planets hardly survive the RGB tip and the early AGB phases. Based on observations collected at the La Silla Observatory, ESO (Chile), with HARPS/3.6 m (runs ID 075.C-0140, 076.C-0429, 077.C-0088, and 078.C-0133) and with UVES/VLT at the Cerro Paranal Observatory (run 079.C-0131).