Abstract The heat transfer and entropy generation in a magnetohydrodynamic flow of Al 2 O 3 –water nanofluid through a porous vertical microchannel with nonlinear radiative heat flux were investigated numerically. Then, combined effects of nanoparticle volume fraction, hydrodynamic slip, magnetic field, suction/injection and thermal radiation on heat transfer and entropy generation were studied. The dimensionless governing equations were solved numerically by applying Runge–Kutta integration method together with shooting technique. In this study, the accuracy of the numerical results was verified by comparing its predictions with exact solutions of model without both radiation effects and buoyancy force. Here, different from previous literature, heat transfer subject to nonlinear thermal radiation, Joule heating and viscous dissipation was solved and analyzed using conjugate convective-radiative heat transfer on the boundary surfaces. Moreover, influences of pertinent parameters on nanofluid velocity, temperature, local and global entropy generation and Nusselt number were discussed in detail and illustrated graphically. Based on the numerical results, it was proved that the global entropy generation decreased with both nanoparticle volume fraction and suction/injection Reynolds number while it increased with Grashof number (Buoyancy force intensity), radiation parameter and conduction-radiation parameter. In addition, it was possible to determine optimum values of slip flow with minimum values of global entropy generation rate. The Nusselt number was also calculated and explored for different conditions. In this way, optimum values of Grashof number with maximum heat transfer on the heated left plate were derived.