We demonstrate a scheme for nonlinear pulse compression at high average powers based on self-phase modulation in a multi-pass cell using fused silica as the nonlinear medium. The scheme is suitable for compression of pulses with peak powers exceeding the threshold for critical self-focusing. At >400 W of input power, the pulses of a Yb:YAG-Innoslab laser system (10 MHz repetition rate, 850 fs pulse duration) are spectrally broadened from 1.6 to >13.5 nm bandwidth while maintaining almost diffraction-limited beam quality. The chirp is removed with a dispersive mirror compressor, and pulse durations of 170 fs at an output power of 375 W are achieved. The compression unit reaches an overall transmission of >90%.