The goal of this paper is to investigate the lithium stellar content of Milky Way stars in order to put constraints on the lithium chemical enrichment in our Galaxy, in particular in both the thin and thick discs. We show that the interstellar lithium abundance increases with metallicity by 1dex from [M/H]=-1dex to +0.0dex. Moreover, we find that this lithium ISM abundance decreases by about 0.5dex at super-solar metallicity. Based on a chemical separation, we also observed that the stellar lithium content in the thick disc increases rather slightly with metallicity while the thin disc shows a steeper increase. The lithium abundance distribution of alpha-rich metal-rich stars has a peak at A(Li)~3 dex. We conclude that the thick disc stars suffered of a low lithium chemical enrichment, showing lithium abundances rather close to the Spite plateau while the thin disc stars clearly show an increasing lithium chemical enrichment with the metallicity, probably thanks to the contribution of low-mass stars.