Low temperatre X-ray diffraction analysis of Nb3Sn multifilamentary wires produced by the bronze route has been performed at different orientations. An important volume fraction of the Nb3Sn filaments is found to undergo a stress-induced cubic-tetragonal transformation with at 10 K. Strong texturing of the tetragonal phase has been detected: the shorter c axis is preferentially oriented along the wire axis. The present data furnish a direct indication for the uniaxial stress distribution in the wire. Careful Hc2(T) measurements on a wire at the vicinity of Tcreveal a sharp change of the slope dHc2/dT|Tc. This is interpreted as a physical proof for the coexistence of two phases at low temperature, with Tc(cub.) < Tc(tetr.), but Hc2(cub.) > Hc2(tetr.). Based on these data, the variation of the ratio Jcm/Jcoof Nb3Sn multifilamentary wires under uniaxial stress can be correlated with the presence of the tetragonal phase. The applied strain is believed to affect as well the the tetragonal volume fraction as the ratio c/a of the tetragonal phase.