We calculated the electronic structure of a vortex in a pseudogapped superconductor within a model featuring strong correlations. With increasing strength of the correlations, the BCS core states are suppressed and the spectra inside and outside the core become similar. If the correlations are short range, we find new core states in agreement with the observations in YBa2Cu3O7-δ and Bi2Sr2CaCu2O8+δ. Our results point to a common phenomenology for these two systems and indicate that normal-state correlations survive below Tc without taking part in the overall phase coherence.