Poly(ethylene-co-vinyl acetate)-clay nanocomposites containing different filler loading were prepared. Transport of gases through the composite membranes was investigated, and the results were compared with the unfilled one. Transport studies revealed that gas transport was considerably reduced by the incorporation of clay filler into the polymer matrix especially at a loading of 3 wt % filler. It was also found that permeability increased for composites containing more than 3 wt %. This is presumably due to aggregation of clay filler at higher loading. The morphology of nanocomposites was analyzed by X-ray diffraction and transmission electron microscopy. Dielectric loss data were also used to understand the uniformity of dispersion of nanoparticles. The effect of free volume on transport behavior was investigated by positron lifetime spectroscopic analysis. The cross-link density values were estimated to correlate with the gas barrier properties. The oxygen/nitrogen selectivity of these membranes was investigated.