Asteroid (4) Vesta is known as the unique large object of the main belt showing a basaltic crust and is a target of the NASA Dawn Mission. Vesta composition is also similar to that of basaltic achondrite meteorites, specifically the Eucrites, Diogenites and Howardites (HED). On the other hand, in recent taxonomies, asteroids showing a spectrum similar to that of Vesta have been classified as V-type. The spectroscopic link between the V-type asteroids in the vicinity of Vesta, in near-Earth orbits, and the HED meteorites seems to be quite consistent, especially if we take into account that basaltic material is very rare in the asteroid belt. The depth, width and spectral placement of the 1- and 2-micron mafic silicate absorption features in the reflectance spectrum of Vesta and V-type asteroids help to characterize the surface mineralogy of these objects. We present reflectance spectra of 19 V-type asteroids obtained at the 3.6 m Telescopio Nazionale Galileo covering 0.8 to 2.5 microns. For 8 of these asteroids we obtained also visible spectra in the same observational run. The range from 0.8 to 2.5 microns, encompassing the 1 and 2 microns pyroxene features, allows a precise mineralogical characterization of these asteroids. Our results on the mineralogy of V-type asteroids in the neighbourhood of Vesta corroborate previous works about the existence of different kinds of basalts at the same time that do not show any clear correlation between mineralogies and the objects being, or not, members of the Vesta's dynamical family. The obtained data suggests the possible coexistence of distinct mineralogical groups among the V-type asteroids, either probing differents layers of (4) Vesta or coming from different bodies. "Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Centro Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatório del Roque de los Muchachos of the Instituto de Astrofísica de Canarias"