It was recently shown that nonseparable density operators on the Hilbert space H₁⊗H₂ are trace norm dense if either factor space has infinite dimension. We show here that nonlocal states, i.e., states whose correlations cannot be reproduced by any local hidden variable model, are also dense. Our constructions distinguish between the case dim H₁ = dim H₂ = ∞, where we show that states violating the Clauser-Horne-Shimony-Holt (CHSH) inequality are dense, and the case dim H₁<dimH₂=∞, where we identify open neighborhoods of nonseparable states that do not violate the CHSH inequality but show that states with a subtler form of nonlocality (often called "hidden" nonlocality) remain dense.