A numerical analysis was performed to study the effects of variable viscosity and thermal conductivity under non-uniform wall boundary conditions over a vertical plate embedded in a non-Darcy porous medium in the presence of chemical reaction. Coupled heat and mass transfer of the natural convective boundary layer with viscous nanofluid are considered. Further, the viscosity and thermal conductivity are assumed as the function of the nanoparticle volume fraction, while the Darcy-Forchheimer model is used for the porous medium. Using the Oberbeck-Boussinesq and standard boundary layer approximations, the governing equations describing the steady-state conservation of mass, momentum, energy as well as conservation of nanoparticles for nanofluids in the presence of chemical reaction parameter are transformed into the set of ordinary differential equations with similarity variables. These equations together with associated boundary conditions are solved numerically by using the finite-difference method. To validate the numerical results, comparison is made with those available and a good agreement is obtained. Many numerical results are displayed graphically to illustrate the influence of the various parameters of interest on different profiles. Extensive numerical investigations show that the flow field, temperature and nanoparticle volume fraction shapes are significantly influenced by variable viscosity and thermal conductivity parameter, variable thermal conductivity parameter, non-Darcy parameter, reaction rate parameter and Convective heat transfer coefficient. Furthermore, the dependency of the Nusselt and Sherwood numbers on these parameters is investigated.