(4) Vesta is the unique large asteroid with a basaltic surface present in the Asteroid Belt. In recent years, several smaller asteroids having similar composition have been discovered in the neighbourhood of Vesta and even far away from it. It has been suggested that all these small asteroids came from Vesta even if dynamical studies do not fully corroborate this scenario. A better insight on this problem can be obtained through a precise mineralogical characterization of some of these bodies. In the present work we obtained low resolution VNIR spectra of V-type asteroids in the neighbourhood of Vesta. The observations were obtained on April 22 and 23, 2003, with the 3.6m Telescopio Nazionale Galileo (TNG) using NICS and the Amici disperser. This instrumentation yields a complete 0.9 - 2.5mm spectrum. Visible spectra, in the range of 0.4 - 1.0mm, were also obtained, on April 24, for some of the asteroids using DOLORES Low Resolution Spectrograph. All the data has been reduced following standard procedures with the IRAF package. Near-infrared and visible spectra were obtained for 18 and 8 asteroids, respectively. Among the observed asteroids 12 are members of the Vesta's dynamical family while the remaining are V-type asteroids in the neighbourhood, but not members of the family. The precise mineralogical composition of the observed asteroids was determined using the technique described in Gaffey et al.[Icarus 196, 573, 1993]. The compositional analysis of several V-type asteroids in the neighbourhood of Vesta corroborates previous works about the existence of different kinds of basalts. On the other side, there seem to be no clear correlation between the mineralogy and the objects being, or not, members of the dynamical family of Vesta. "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 Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias."