Dawn space mission will provide the first, detailed data of two of the major bodies in the main asteroid belt, Vesta and Ceres. Through its connection with Howardite, Eucrite, Diogenite (HED) meteorites, Vesta is known as one of the first bodies to have accreted and differentiated in the solar nebula, predating the formation of Jupiter and surviving the violent evolution of the early Solar system. The formation time of Ceres instead is unknown, but it should not postdate that of Jupiter by far, since the perturbations of the giant planet stopped planetary accretion in the main asteroid belt. In this work we modelled the collisional histories of Vesta and Ceres at the time of the formation of Jupiter, assumed to be the first giant planet to form. In this first investigation of the evolution of the early Solar system, we did not include the presence of planetary embryos in the disc of planetesimals but concentrated on the role of the forming Jupiter and the effects of its possible inward migration due to disc–planet interactions. Our results clearly indicate that the formation of the giant planet caused an intense early bombardment in the orbital region of the main asteroid belt. We explored the effects of such bombardment on Vesta and Ceres assuming different size distributions of the primordial planetesimals. According to our results, Vesta and Ceres would not have survived the Jovian early bombardment if the disc was populated mainly by large planetesimals like those predicted to form in turbulent circumstellar discs. Discs dominated by small bodies, like those predicted to form in quiescent circumstellar discs, or with a varying fraction of the mass in the form of larger (D≥ 100 km) planetesimals, represent more favourable environments for the survival of the two asteroids. The abundance of planetesimals, especially the larger ones, proved a critical factor in this regard. The extent of Jupiter’s radial migration due to disc–planet interactions proved itself another critical factor. In those scenarios where they survive, both asteroids had their surfaces saturated by craters as big as 150 km and a few as big as 200–300 km. In the case of Vesta, the Jovian early bombardment would have significantly eroded (locally or globally) the crust and possibly caused effusive phenomena similar to the lunar maria, whose crystallization time would then be directly linked to the time of the formation of Jupiter.