We proposed and experimentally demonstrated a technique for the suppression of unwanted modes in double-clad fibers with a high core-to-clad diameter ratio by introducing high-index absorbing inclusions into the first cladding of the fibers. These inclusions disturb the shape of undesirable modes, and a noticeable part of the power becomes localized inside the inclusion, resulting in an increase in the propagation loss of these modes. Two fiber designs were studied and realized: one with cylindrical symmetry and an absorbing high-index ring as the inclusion and another with high-index absorbing rods inserted around the fiber core. In both cases, the possibility of achieving perfect single-mode propagation was demonstrated both theoretically and experimentally.