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We report on tunneling measurements that reveal the evolution of the quasiparticle state density in two rare earth perovskite nickelates, NdNiO3 and LaNiO3, that are close to a bandwidth controlled metal to insulator transition. We measure the opening of a sharp gap of ∼30 meV in NdNiO3 in its insulating ground state. LaNiO3, which remains a correlated metal at all practical temperatures, exhibits a pseudogap of the same order. The results point to both types of gaps arising from a common origin, namely, a quantum critical point associated with the T = 0 K metal-insulator transition. The results support theoretical models of the quantum phase transition in terms of spin and charge instabilities of an itinerant Fermi surface.