The recurrent outbursts that characterize low-mass binary systems reflect thermal state changes in their associated accretion discs. The observed outbursts are connected to the strong variation in disc opacity as hydrogen ionizes near 5000 K. This physics leads to accretion disc models that exhibit bistability and thermal limit cycles, whereby the disc jumps between a family of cool and low-accreting states and a family of hot and efficiently accreting states. Previous models have parametrized the disc turbulence via an alpha (or ‘eddy’) viscosity. In this paper we treat the turbulence more realistically via a suite of numerical simulations of the magnetorotational instability (MRI) in local geometry. Radiative cooling is included via a simple but physically motivated prescription. We show the existence of bistable equilibria and thus the prospect of thermal limit cycles, and in so doing demonstrate that MRI-induced turbulence is compatible with the classical theory. Our simulations also show that the turbulent stress and pressure perturbations are only weakly dependent on each other on orbital times; as a consequence, thermal instability connected to variations in turbulent heating (as opposed to radiative cooling) is unlikely to operate, in agreement with previous numerical results. Our work presents a first step towards unifying simulations of full magnetohydrodynamic turbulence with the correct thermal and radiative physics of the outbursting discs associated with dwarf novae, low-mass X-ray binaries and possibly young stellar objects.