ABSTRACT The Event Horizon Telescope (EHT) collaboration has produced the first resolved images of the supermassive black holes at the centre of our galaxy and at the centre of the elliptical galaxy M87. As both technology and analysis pipelines improve, it will soon become possible to produce spectral index maps of black hole accretion flows on event horizon scales. In this work, we predict spectral index maps of both M87* and Sgr A* by applying the general relativistic radiative transfer (GRRT) code ipole to a suite of general relativistic magnetohydrodynamic (GRMHD) simulations. We analytically show that the spectral index increases with increasing magnetic field strength, electron temperature, and optical depth. Consequently, spectral index maps grow more negative with increasing radius in almost all models, since all of these quantities tend to be maximized near the event horizon. Additionally, photon ring geodesics exhibit more positive spectral indices, since they sample the innermost regions of the accretion flow with the most extreme plasma conditions. Spectral index maps are sensitive to highly uncertain plasma heating prescriptions (the electron temperature and distribution function). However, if our understanding of these aspects of plasma physics can be tightened, even the spatially unresolved spectral index around 230 GHz can be used to discriminate between models. In particular, Standard and Normal Evolution (SANE) flows tend to exhibit more negative spectral indices than Magnetically Arrested Disc (MAD) flows due to differences in the characteristic magnetic field strength and temperature of emitting plasma.