Energetic neutral atoms (ENAs) are produced by the neutralization of energetic ions formed by shock-accelerated gradual solar energetic particle events (SEP). These high-energy ENAs (HENAs) can reach the Earth earlier than the associated SEPs and provide information about the SEPs at the lower corona. The HENA properties observed at Earth depend on the properties of the coronal mass ejection (CME)-driven shocks that accelerate the SEPs. With a model of HENA production in a shock-accelerated SEP event, we semi-quantitatively investigate the energy-time spectrum of HENAs depending on the width, propagation speed, and direction of the shock, as well as the density and ion abundances of the lower corona. Compared to the baseline model parameters, the cases with a wider shock width angle or a higher coronal density would increase the HENA flux observed at the Earth, while the case with an Earth-propagating shock shows a softened HENA spectrum. The comparison of expected HENA fluxes in different cases with a flight-proven ENA instrument suggests that solar HENAs can feasibly be monitored with current technologies, which could provide a lead time of 2-3 hours for SEPs at a few MeV. We propose that monitoring of solar HENAs could provide a new method to forecast shock-driven SEP events that could have significant space weather impacts on the near-Earth environment.