Computational auditory models are important tools for gaining new insights into hearing mechanisms, and they can provide a foundation for bio-inspired speech and audio processing algorithms. However, accurate models often entail an immense computational effort, rendering their application unfeasible if quick execution is required. This paper presents a WaveNet-based approximation of the normal-hearing cochlear filtering and inner hair cell (IHC) transduction stages of a widely used auditory model [Zilany and Bruce (2006). J. Acoust. Soc. Am. 120(3), 1446–1466]. The WaveNet model was trained and optimized using a large dataset of clean speech, noisy speech, and music for a wide range of sound pressure levels (SPLs) and characteristic frequencies between 125 Hz and 8 kHz. The model was evaluated with unseen (noisy) speech, music signals, sine tones, and click signals at SPLs between 30 and 100 dB. It provides accurate predictions of the IHC receptor potentials for a given input stimulus and allows an efficient execution with processing times up to 250 times lower compared to an already optimized reference implementation of the original auditory model. The WaveNet model is fully differentiable, thus, allowing its application in the context of deep-learning-based speech and audio enhancement algorithms.