The stress-induced martensitic transformation in tensioned nickel-titanium shape-memory alloys proceeds by propagation of macroscopic fronts of localized deformation. We used three-dimensional synchrotron x-ray diffraction to image at micrometer-scale resolution the grain-resolved elastic strains and stresses in austenite around one such front in a prestrained nickel-titanium wire. We found that the local stresses in austenite grains are modified ahead of the nose cone-shaped buried interface where the martensitic transformation begins. Elevated shear stresses at the cone interface explain why the martensitic transformation proceeds in a localized manner. We established the crossover from stresses in individual grains to a continuum macroscopic internal stress field in the wire and rationalized the experimentally observed internal stress field and the topology of the macroscopic front by means of finite element simulations of the localized deformation.