Temporal modulations in the envelope of acoustic waveforms at rates around 4 Hz constitute a strong acoustic cue in speech and other natural sounds. It is often assumed that the ascending auditory pathway is increasingly sensitive to slow amplitude modulation (AM), but sensitivity to AM is typically considered separately for individual stages of the auditory system. Here, we used blood oxygen level dependent (BOLD) fMRI in twenty human subjects (10 male) to measure sensitivity of regional neural activity in the auditory system to 4 Hz temporal modulations. Participants were exposed to AM noise stimuli varying parametrically in modulation depth to characterize modulation-depth effects on BOLD responses. A Bayesian hierarchical modeling approach was used to model potentially nonlinear relations between AM depth and group-level BOLD responses in auditory regions of interest (ROIs). Sound stimulation activated the auditory brainstem and cortex structures in single subjects. BOLD responses to noise exposure in core and belt auditory cortices scaled positively with modulation depth. This finding was corroborated by whole-brain cluster-level inference. Sensitivity to AM depth variations was particularly pronounced in the Heschl's gyrus but also found in higher-order auditory cortical regions. None of the sound-responsive subcortical auditory structures showed a BOLD response profile that reflected the parametric variation in AM depth. The results are compatible with the notion that early auditory cortical regions play a key role in processing low-rate modulation content of sounds in the human auditory system.