A study of the existence of non-nuclear maxima (NNM) in the electron density of simple metals is presented. The calculations are based on density functional theory and the linearized augmented plane wave method in which the basis set quality can be systematically improved. We find that only lithium and beryllium exhibit NNM at calculated and experimental equilibrium volumes. The existence of NNM is found to be caused by a relatively small extension of their core densities. The NNM appear in both the local density and generalized gradient approximations. The use of the latter favors the existence of NNM and at the same time improves the agreement with measured structure factors and lattice constants. Therefore, we argue that it is very likely that NNM would also appear for the exact functional.