In this article, the three-dimensional Magnetohydrodynamics flow of a nanofluid over a horizontal non-linearly stretching sheet in bilateral directions under boundary layer approximation is addressed. A two-phase model has been used for the nanofluid. The influences of thermophoresis, Brownian motion and thermal radiation on heat and mass transfers are considered. Two different cases for the heat and mass transfers are studied. In the first case, uniform wall temperature and zero nanoparticles flux due to thermophoresis are considered. In the second case, prescribed heat and mass fluxes at the boundary are considered. By using the appropriate transformations, a system of non-linear partial differential equations along with the boundary conditions is transformed into coupled non-linear ordinary differential equations. Numerical solutions of the self-similar equations are obtained using a Runge–Kutta method with a shooting technique. Our results for special cases are compared with the available results in the literature, and the results are found to be in good agreement. It is observed that the pertaining parameters have significant effects on the characteristics of flow, heat and mass transfer. The results are presented and discussed in detail through illustrations.