Multimode optical fibers are attracting a growing interest for their capability to transport high-power laser beams, coupled with novel nonlinear optics-based applications. However, optical fiber breakdown occurs when beam intensities exceed a certain critical value. Optical breakdown associated with irreversible modifications of the refractive index, triggered by multiphoton absorption, has been largely exploited for fiber material micro-structuration. Here we show that, for light beam intensities slightly below the breakdown threshold, nonlinear absorption strongly affects the dynamics of a propagating beam as well. We experimentally analyze this subthreshold regime and highlight the key role played by spatial self-imaging in graded-index fibers for enhancing nonlinear optical losses. We characterize the nonlinear power transmission properties of multimode fibers for femtosecond pulses propagating in the near-infrared spectral range. We show that an effective N-photon absorption analytical model is able to describe the experimental data well.