We report the generation of a broadband supercontinuum (SC) from 790 to 2900 nm in a tellurite graded-index (GRIN) multimode fiber with a nanostructured core. We study the SC dynamics in different dispersion regimes and observe near-single-mode spatial intensity distribution at high input energy values. Numerical simulations of the (3 + 1)D generalized nonlinear Schrödinger equation are in good agreement with our experiments. Our results open a new avenue for the generation of high-power mid-infrared SC sources in soft-glass fibers.