This article proposes a numerical strategy to simulate the mass transport by surface and lattice diffusion into a granular packing. This strategy is based on two cornerstones. First, the developed approach is based on a Eulerian description of the problem: the grains are described by using a Level-Set function, and can evolve through a fixed mesh, with respect to the physical laws. In this way, the mesh does not experience large distortions and topological changes, such as the formation of necks or of closed porosity, are implicitly taken into account by the Level-Set method. Second, the computation of the mechanical state into the grains is directly performed when considering the lattice diffusion route. Hence, a mechanical problem, coupling the grain elastic behavior to the fluid behavior of the surrounding phase, is established and solved by finite element. The diffusion flux is then related to the gradient of the pressure field. The results obtained with this numerical strategy are compared with success to the usual geometrical models for two spherical grains. The possibilities of the numerical approach are shown by presenting the changes occurring by lattice diffusion into a granular packing.