The Helioseismic and Magnetic Imager onboard the Solar Dynamics Observatory has provided unique observations of off-limb flare emission. White-light (WL) continuum enhancements were detected in the continuum channel of the Fe 6173 A line during the impulsive phase of the observed flares. In this paper we aim to determine which radiation mechanism is responsible for such an enhancement being seen above the limb, at chromospheric heights around or below 1000 km. Using a simple analytical approach, we compare two candidate mechanisms, the hydrogen recombination continuum (Paschen) and the Thomson continuum due to scattering of disk radiation on flare electrons. Both mechanisms depend on the electron density, which is typically enhanced during the impulsive phase of a flare as the result of collisional ionization (both thermal and also non-thermal due to electron beams). We conclude that for electron densities higher than $10^{12}$ cm$^{-3}$, the Paschen recombination continuum significantly dominates the Thomson scattering continuum and there is some contribution from the hydrogen free-free emission. This is further supported by detailed radiation-hydrodynamical (RHD) simulations of the flare chromosphere heated by the electron beams. We use the RHD code FLARIX to compute the temporal evolution of the flare heating in a semi-circular loop. The synthesized continuum structure above the limb resembles the off-limb flare structures detected by HMI, namely their height above the limb, as well as the radiation intensity. These results are consistent with recent findings related to hydrogen Balmer continuum enhancements, which were clearly detected in disk flares by the IRIS near-ultraviolet spectrometer.
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