The tumbling to tank-treading (TB-TT) transition for red blood cells (RBCs) has been widely investigated, with a main focus on the effects of the viscosity ratio λ (i.e., the ratio between the viscosities of the fluids inside and outside the membrane) and the shear rate γ ̇ applied to the RBC. However, the membrane viscosity μm plays a major role in a realistic description of RBC’s dynamics, and only a few works have systematically focused on its effects on the TB-TT transition. In this work, we provide a parametric investigation on the effect of membrane viscosity μm on the TB- TT transition, for a single RBC. It is found that, at fixed viscosity ratios λ, larger values of μm lead to an increased range of values of capillary number at which the TB-TT transition occurs. We systematically quantify such an increase by means of mesoscale numerical simulations based on the lattice Boltzmann models.