Neural processing along the ascending auditory pathway is often associated with a progressive reduction in characteristic processing rates. For instance, the well-known frequency-following response (FFR) of the auditory midbrain, as measured with electroencephalography (EEG), is dominated by frequencies from ∼100 Hz to several hundred Hz, phase-locking to the acoustic stimulus at those frequencies. In contrast, cortical responses, whether measured by EEG or magnetoencephalography (MEG), are typically characterized by frequencies of a few Hz to a few tens of Hz, time-locking to acoustic envelope features. In this study we investigated a crossover case, cortically generated responses time-locked to continuous speech features at FFR-like rates. Using MEG, we analyzed responses in the high gamma range of 70–200 Hz to continuous speech using neural source-localized reverse correlation and the corresponding temporal response functions (TRFs). Continuous speech stimuli were presented to 40 subjects (17 younger, 23 older adults) with clinically normal hearing and their MEG responses were analyzed in the 70–200 Hz band. Consistent with the relative insensitivity of MEG to many subcortical structures, the spatiotemporal profile of these response components indicated a cortical origin with ∼40 ms peak latency and a right hemisphere bias. TRF analysis was performed using two separate aspects of the speech stimuli: a) the 70–200 Hz carrier of the speech, and b) the 70–200 Hz temporal modulations in the spectral envelope of the speech stimulus. The response was dominantly driven by the envelope modulation, with a much weaker contribution from the carrier. Age-related differences were also analyzed to investigate a reversal previously seen along the ascending auditory pathway, whereby older listeners show weaker midbrain FFR responses than younger listeners, but, paradoxically, have stronger cortical low frequency responses. In contrast to both these earlier results, this study did not find clear age-related differences in high gamma cortical responses to continuous speech. Cortical responses at FFR-like frequencies shared some properties with midbrain responses at the same frequencies and with cortical responses at much lower frequencies.