Aims. Exoplanet-host stars (EHS) are known to present superficial chemical abundances different from those of stars without any detected planet (NEHS). EHS are, on the average, overmetallic compared to the Sun. The observations also show that, for cool stars, lithium is more depleted in EHS than in NEHS. The aim of this paper is to obtain constraints on possible models able to explain this difference, in the framework of overmetallic models compared to models with solar abundances. Methods. We have computed main sequence stellar models with various masses and metallicities. The results show different behaviour for the lithium destruction according to those parameters. We compare these results to the spectroscopic observations of lithium. Results. Our models show that the observed lithium differences between EHS and NEHS are not directly due to the overmetallicity of the EHS: some extra mixing is needed below the convective zones. We discuss possible explanations for the needed extra mixing, in particular an increase of the mixing efficiency associated with the development of shear instabilities below the convective zone, triggered by angular momentum transfer due to the planetary migration.
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