We present kinetic Monte Carlo simulations of growing nanoclusters on a weakly interacting substrate within an fcc binary alloy model. This model is designed to describe essential structural and magnetic properties of CoPt 3 -type clusters grown by molecular beam epitaxy. Due to a growth-induced structural anisotropy located near the cluster surface, these clusters exhibit a perpendicular magnetic anisotropy in a certain temperature range of growth, in agreement with experiments. By applying an external magnetic field during the simulated growth, a clear numerical evidence for an enhanced perpendicular magnetic anisotropy in these systems is found under certain growth conditions.