The Levenberg-Marquardt method (LM) is widely used for solving nonlinear systems of equations, as well as nonlinear least-squares prob- lems. In this paper, we consider local convergence issues of the LM method when applied to nonzero-residue nonlinear least-squares problems under an error bound condition, which is weaker than requiring full-rank of the Jacobian in a neighborhood of a stationary point. Differently from the zero-residue case, the choice of the LM parameter is shown to be dic- tated by (i) the behavior of the rank of the Jacobian, and (ii) a combined measure of nonlinearity and residue size in a neighborhood of the set of (possibly non-isolated) stationary points of the sum of squares function.