Self-supervised language models are very effective at predicting high-level cortical responses during language comprehension. However, the best current models of lower-level auditory processing in the human brain rely on either hand-constructed acoustic filters or representations from supervised audio neural networks. In this work, we capitalize on the progress of self-supervised speech representation learning (SSL) to create new stateof-the-art models of the human auditory system. Compared against acoustic baselines, phonemic features, and supervised models, representations from the middle layers of self-supervised models (APC, wav2vec, wav2vec 2.0, and HuBERT) consistently yield the best prediction performance for fMRI recordings within the auditory cortex (AC). Brain areas involved in low-level auditory processing exhibit a preference for earlier SSL model layers, whereas higher-level semantic areas prefer later layers. We show that these trends are due to the models' ability to encode information at multiple linguistic levels (acoustic, phonetic, and lexical) along their representation depth. Overall, these results show that self-supervised models effectively capture the hierarchy of information relevant to different stages of speech processing in human cortex.