Context. The diffusion of volatile species on amorphous solid water ice affects the chemistry on dust grains in the interstellar medium as well as the trapping of gases enriching planetary atmospheres or present in cometary material. Aims. The aim of the work is to provide diffusion coefficients of CH4 on amorphous solid water (ASW) and to understand how they are affected by the ASW structure. Methods. Ice mixtures of H2O and CH4 were grown in different conditions and the sublimation of CH4 was monitored via infrared spectroscopy or via the mass loss of a cryogenic quartz crystal microbalance. Diffusion coefficients were obtained from the experimental data assuming the systems obey Fick's law of diffusion. Monte Carlo simulations were used to model the different amorphous solid water ice structures investigated and were used to reproduce and interpret the experimental results. Results. Diffusion coefficients of methane on amorphous solid water have been measured to be between 10-12 and 10-13 cm2 s-1 for temperatures ranging between 42 K and 60 K. We show that diffusion can differ by one order of magnitude depending on the morphology of amorphous solid water. The porosity within water ice and the network created by pore coalescence enhance the diffusion of species within the pores. The diffusion rates derived experimentally cannot be used in our Monte Carlo simulations to reproduce the measurements. Conclusions. We conclude that Fick's laws can be used to describe diffusion at the macroscopic scale, while Monte Carlo simulations describe the microscopic scale where trapping of species in the ices (and their movement) is considered.