Molecular dynamics simulations were carried out to study the structural and transport properties of carbon dioxide, methane, and their mixture at 298.15 K in Na-montmorillonite clay in the presence of water. The simulations show that, the self-diffusion coefficients of pure CO2 and CH4 molecules in the interlayers of Na-montmorillonite decrease as their loading increases, possibly because of steric hindrance. The diffusion of CO2 in the interlayers of Na-montmorillonite, at constant loading of CO2, is not significantly affected by CH4 for the investigated CO2/CH4 mixture compositions. We attribute this to the preferential adsorption of CO2 over CH4 in Na-montmorillonite. While the presence of adsorbed CO2 molecules, at constant loading of CH4, very significantly reduces the self-diffusion coefficients of CH4, and relatively larger decrease in those diffusion coefficients are obtained at higher loadings. The preferential adsorption of CO2 molecules to the clay surface screens those possible attractive surface sites for CH4. The competition between screening and steric effects leads to a very slight decrease in the diffusion coefficients of CH4 molecules at low CO2 loadings. The steric hindrance effect, however, becomes much more significant at higher CO2 loadings and the diffusion coefficients of methane molecules significantly decrease. Our simulations also indicate that, similar effects of water on both carbon dioxide and methane, increase with increasing water concentration, at constant loadings of CO2 and CH4 in the interlayers of Na-montmorillonite. Our results could be useful, because of the significance of shale gas exploitation and carbon dioxide storage.