The Eurybates family is a compact core inside the Menelaus clan, located in the L4 swarm of Jupiter Trojans. Fornasier et al. (2007) found that this family exhibits a peculiar abundance of spectrally flat objects, similar to Chiron-like Centaurs and C-type main belt asteroids. On the basis of the visible spectra available in literature, Eurybates familys members seemed to be good candidates for having on their surfaces water/water ice or aqueous altered materials. To improve our knowledge of the surface composition of this peculiar family, we carried out an observational campaign at the Telescopio Nazionale Galileo (TNG), obtaining near-infrared spectra of 7 members. Our data show a surprisingly absence of any spectral feature referable to the presence of water, ices or aqueous altered materials on the surface of the observed objects. Models of the surface composition are attempted, evidencing that amorphous carbon seems to dominate the surface composition of the observed bodies and some amount of silicates (olivine) could be present.