The martian surface is predominantly covered by FeO-rich basalts and their alteration products. Several samples, either analyzed in situ by rovers or recovered as meteorites, might represent primitive (i.e. near-primary) basaltic melts that can shed light on the mineralogy, the bulk composition, and the temperature of their mantle sources. We recently developed a new melting model, called MAGMARS, that can predict the melt compositions of FeO-rich mantles and the martian mantle in particular (Collinet et al., submitted to JGR:P). It represents a more accurate alternative to pMELTS (Ghiorso et al., 2002, G3), which systematically overestimates the FeO and MgO content of martian melts and underestimates the SiO2 content (by up to 8 wt.%). MAGMARS can simulate near-fractional and batch melting of various mantle compositions. For example, MAGMARS can produce melts identical to the Adirondack-class basalts by near-fractional melting, between 2.3 and 1.7 GPa, of a depleted mantle with a potential temperature (Tp) of 1390°C (~7 wt.% melt fraction). For this study, MAGMARS is applied to all other martian basalts from which the primary melt compositions can be inferred in order to constrain their mantle sources: the Columbia hills basalts, igneous rocks from Gale crater, shergottites, nakhlites and Northwest Africa (NWA) 7034/7533. We find that a few basaltic clasts in the pre-Noachian polymict regolith breccia NWA 7034/7533 are the only samples with bulk compositions that could represent melts derived from a primitive mantle. The Columbia hills basalts (Gusev crater), alkali-rich rocks from Gale crater, nakhlites and enriched shergottites are most easily reproduced by melting depleted mantle reservoirs that were re-fertilized to different degrees in alkalis by fluids or melts (i.e. metasomatized sources). Most martian basalts, with the exception of depleted shergottites, can be produced from martian mantle reservoirs with Mg# comprised between 75 and 81. From this sample set, the melting conditions of the martian mantle seem to remain relatively stable through time (Tp = 1400 ± 100 ºC and P = 2 ± 0.5 GPa) but the depleted nature of all mantle sources sampled after the pre-Noachian points towards an early crust-mantle differentiation.