Aluminium is the majority element in many quasicrystals and expected to be the most mobile element, but its diffusion properties are hardly accessible to experiment. Here we investigate aluminum diffusion in decagonal Al-Ni-Co and Al-Cu-Co quasicrystals by molecular dynamics simulations, using classical effective pair potentials. Above two-thirds of the melting temperature, strong aluminum diffusion is observed. The diffusion constant is measured as a function of temperature and pressure, from which the activation enthalpies and activation volumes are determined. As there are no vacancies in the samples, the diffusion, which is anisotropic, must use a direct mechanism. The high mobility of aluminium is also relevant for structure determination, and will contribute to diffuse scattering. The qualitative behavior of the dynamics is confirmed by ab initio simulations.