We investigate the effects of the convection in the hyperaccretion disc around a stellar-mass black hole, which is considered to be the central engine of gamma-ray bursts (GRBs), with simple analytical calculations. If the convective energy transfer in the vertical direction becomes efficient compared with the inward advective energy transport, the hyperaccretion disc is expected to be hotter and the neutrino emission due to the electron–positron annihilation becomes the most efficient cooling process. We find that the sequence of the thermal equilibrium solutions for the convective hyperaccretion disc would have a viscously unstable branch, especially when the viscosity parameter is relatively small (α≲ 0.01). This means that the sporadical mass accretion on to a black hole would occur in this disc. We propose that this process can be the origin of the highly variable light curves observed in the prompt emissions of GRBs.