A density functional approach based on Kierlik-Rosinberg theory is applied to a two-component Lennard-Jones mixture in contact with a selectively permeable membrane. The results of theoretical calculations are compared with the grand canonical ensemble Monte Carlo simulations. The density profiles obtained for temperatures higher than the critical temperature agree well with simulation data if the difference in the diameters of particles of two components is small. We have also obtained that the agreement between the theory and simulations is reasonable if the particles of the component that penetrates the membrane are twice larger than the particles of nonpenetrating component. For other diameter ratios and for the case when the density of the mixture is high, the theory disagrees with simulations. At temperatures lower than the critical temperature the theoretical density profiles also have been evaluated, and a possibility of multiple solutions of the density profile equation, corresponding to the existence of different phases, is discussed.