High resolution neutron inelastic scattering techniques were used to measure tunneling spectra of methane (CH4) in its phase III. By increasing the pressure from 0.6 to 1.8 kbar the tunneling lines shift by about a factor of 1.5. The rather complicated spectra can be equally well described by two simple structural models. Both contain two inequivalent molecular sites of low symmetry. They already have provided a rather good fit of the spectrum of CD4 III. An exponent n = 8±2 of an effective intermolecular potential V∼r−n is obtained by analyzing the pressure dependence of the tunneling frequencies.