The onset of double-diffusive convection in an electrically conducting fluid-saturated porous layer is studied. The convective flow in the porous medium is induced by horizontal temperature and concentration gradients with net horizontal mass throughflow. The effect of magnetic field on the instability of convection is taken into consideration. The stability of the steady-state solution is investigated in two different approaches, namely the linear instability analysis and nonlinear stability analysis. The nonlinear stability analysis is performed by constructing the energy functional. The eigenvalue problems which are derived from the stability analyses are numerically integrated using the shooting and Runge–Kutta methods. The variation in the critical thermal Rayleigh number against each flow governing parameter is shown graphically. It is observed that Hartman number Ha2 delays the onset of convection to commence and helps to reduce the region of subcritical instabilities. When the solute is concentrated at lower boundary of the porous layer, the onset of convection is in the form of stationary modes, but it switches to oscillatory mode of convection when the solute is concentrated at upper boundary. Interestingly, Hartman number Ha2 plays an important role in delaying this transition from stationary mode to oscillatory mode of convection. © 2020, Springer Nature B.V.