We present a new approach for the development of structured optical fibers. It is shown that fibers having an effective gradient index profile with designed refractive index distribution can be developed with internal nanostructuring of the core composed of two glasses. As proof-of-concept, fibers made of two soft glasses with a parabolic gradient index profile are developed. Energy-dispersive X-ray spectroscopy reveals a possibility of selective diffusion of individual chemical ingredients among the sub-wavelength components of the nanostructure. This hints a postulate that core nanostructuring also changes material dispersion of the glasses in the core, potentially opening up unique dispersion shaping possibilities.