To keep massive MIMO systems cost-efficient, power amplifiers with rather small output dynamic ranges are employed. They may distort the transmit signal and degrade the performance. This paper proposes a distortion aware precoding scheme for realistic scenarios in which RF chains have nonlinear characteristics. The proposed scheme utilizes the method of regularized least-squares (RLS) to jointly compensate the channel impacts and the distortion imposed by the RF chains. To construct the designed transmit waveform with low computational complexity, an iterative algorithm based on approximate message passing is developed. This algorithm is shown to track the achievable average signal distortion of the proposed scheme tightly, even for practical system dimensions. The results demonstrate considerable enhancement compared to the state of the art.