High density polyethylene (HDPE)/clay nanocomposites containing nanoclay concentrations of 1, 2.5, and 5 wt% were prepared by a melt blending process. The effects of various types of nanoclays and their concentrations on permeability, solubility, and diffusivity of natural gas in the nanocomposites were investigated. The results were compared with HDPE typically used in the production of liners for the petroleum industry. Four different nanoclays—Cloisite 10A, 15A, 30B and Nanomer 1.44P—were studied in the presence of CH4 and a CO2/CH4 mixture in the temperature range 30–70 °C and pressure range 50–100 bar. The permeability and diffusivity of the gases were considerably reduced by the incorporation of nanoclay into the polymer matrix. Addition of 5 wt% loading of Nanomer 1.44P reduced the permeability by 46% and the diffusion coefficient by 43%. Increasing the pressure from 50 to 100 bar at constant temperature had little influence on the permeability, whereas increasing the temperature from 30 to 70 °C significantly increased the permeability of the gas. Additionally, the effect of crystallinity on permeability, solubility, and diffusivity was investigated. Thus, the permeability of the CO2/CH4 mixture in Nanomer 1.44P nanocomposite was reduced by 47% and diffusion coefficient by 35% at 5 wt% loading, 50 °C, and 100 bar, compared with pure HDPE.