We experimentally studied axial stress distribution in recently developed optical all-solid fibers with nanostructured cores. In this type of fiber, the core is composed of thousands of low and high refractive index glass rods with individual diameters of a few hundred nanometers. A distribution of nanorods determines the effective distribution of the refractive index in the core. A structure of nanorods may introduce unrevealed axial stress distribution after fiber drawing, which may induce change of the expected refractive index value. We studied stress in a custom made nanostructured silica fiber with parabolic refractive index distribution in the core and compared it with the reference SMF-28 fiber. For nanostructured fibers we proved that the axial stress is purely thermal with negligible contribution of mechanical stress. This results in the presence of tensile stress in the fiber core, which is in contrary to a standard telecom fiber, where a compressive stress in the core exists. We showed that measured axial stress has negligible impact on refractive index distribution of nanostructured fibers, thus it does not affect its performance.