The paper is devoted to the problem of approximating reachable sets of a nonlinear control system with state constraints given as a solution set of a nonlinear inequality. A state constraint elimination procedure based on the introduction of an auxiliary constraint-free control system is proposed. The equations of the auxiliary system depend on a small parameter. It is shown that the reachable set of the original system can be approximated in the Hausdorff metric by reachable sets of the auxiliary control system as the small parameter tends to zero. Estimates of the convergence rate are given.