This letter addresses the stability problem of two auxiliary differential equation (ADE) finite-difference time-domain (FDTD) methods for the case of modified Lorentz media, using the combination of the von Neumann method and the Routh-Hurwitz criterion. A rigorous investigation supported by FDTD simulations designates that the stability criterion of the conventional FDTD method can be preserved via the proper selection of the difference and averaging operators. A set of conditions for the dispersive medium parameters is derived, providing the stability limit for both FDTD schemes with practical guidelines for evaluating the fitting of experimentally studied materials.