A study of linear and weakly nonlinear stability analyses of Darcy–Brinkman convection in a water–alumina, nanoliquid‐saturated porous layer for stress‐free isothermal boundaries, when the solid and nanoliquid phases are in local thermal nonequilibrium, is conducted. The critical eigenvalue is found using the Galerkin approach. The effect of the pressure gradient, thermal conductivity ratio, interphase heat transfer coefficient, inverse Darcy number, and Brinkman number on the heat transport and onset of convection is examined and represented graphically. The critical values of wavenumber and nanoliquid Rayleigh number are found for different problem parameter values. The effect of increasing the porosity‐modified ratio of thermal conductivity advances the onset of convection and increases the amount of heat transport, whereas the remaining parameters have the opposite impact on the onset of convection and amount of heat transport. The classical results of the local thermal equilibrium case and Darcy–Bénard convection in the presence of pressure gradient are obtained as a limiting case of the present problem.