In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least squares problems with nonzero residual. We propose a non-stationary Tikhonov method with inexact step computation, specially designed for large-scale problems. At each iteration the method requires the solution of an elliptical trust-region subproblem to compute the step. This task is carried out employing a Lanczos approach, by which an approximated solution is computed. The ad-hoc choice of the trust region radius update and the structure of the step resulting from the use of the Lanczos approach, allows us to prove some regularizing properties of the method. The proposed approach is tested on a parameter identification problem and on an image registration problem, and it is shown to provide important computational savings with respect to its exact counterpart.