In this work, we propose a generalized Levenberg-Marquardt method for nonlinear inverse problems. Compared with the conventional Levenberg-Marqurdt method, the proposed method could be independent of the Fréchet derivative of forward operator F and the iteration points. So it can be used to solve both smooth and non-smooth inverse problems. This method is also designed with general convex penalty terms to detect special features of solutions such as sparsity and piecewise constancy. Convergence analysis of this method is established under a general tangential cone condition (GTCC). In addition, we derive the convergence rate of the proposed method under Hölder-type stability condition. As byproduct, we prove that the general tangential cone condition holds for some PDE inverse problems. Finally, numerical simulations are presented to show the efficiency of the proposed method.