Receptive speech is the ability to understand speech addressed to a person. It is a crucial process for a child’s cognitive development. We examine the relationship between receptive speech and neural tracking of natural speech in 52 children aged 3–8 years to infer the neurophysiological mechanisms underlying speech development. We registered a 32-channel electroencephalogram (EEG) while children listened to narrative audio stories. The temporal response function (TRF) approach was used to study neural tracking features at acoustic and semantic levels. We found a strong positive correlation between the TRF prediction accuracy values that demonstrate the magnitude of neural tracking, and receptive speech abilities measured by the Preschool Language Scales (PLS-5). Topographic analysis of these correlations showed significant clusters of EEG channels in the right temporal region for acoustic tracking, and in the left fronto-central and right parieto-occipital regions for semantic tracking. We assume that these results reflect the development of the brain systems necessary for speech comprehension. To sum up, we suggest that the TRF measures are easy-to-assess neurophysiological markers of receptive speech development in children.