Cell-free (CF) multiuser multiple-input multiple-output (MU-MIMO) systems are an emerging technology that provides service simultaneously to multiple users but suffers from multiuser interference (MUI). In this work, we propose a robust transmit scheme based on rate-splitting (RS) for CF MU-MIMO systems in the presence of imperfect channel state information (CSI) and MUI. We also develop a robust linear precoder design for both private and common precoders based on the minimum mean square error (MMSE) criterion, which incorporates in its design statistical information about the imperfect CSI to provide extra robustness to RS-CF MU-MIMO systems. A statistical analysis is carried out to derive closed-form sum-rate expressions along with a study of the computational complexity of the proposed scheme. Simulation results show that the proposed scheme outperforms conventional robust and non-robust schemes.