The approximate greatest descent (AGD) method and a two-phase AGD method (AGDN) are proposed as new methods for a nonlinear least squares problem. Numerical experiments show that these methods outperform existing methods including the Levenberg-Marquardt method. However, the AGDN method outperforms the AGD method with a faster convergence. If the AGDN method fails due to singularity of the Hessian matrix, the AGD method should be used