Abstract: The design methods of the full-order and the reduced-order observers for discrete-time Lipschitz nonlinear systems with time delay and the sufficient condition for the error convergence are given and proved respectively. The full-order observer solves the two gain matrixes by transforming a matrix inequality with nonlinear terms into two-step linear matrix inequality. However, the gain matrix of observer is obtained easily by solving the linear matrix inequality (LMI), which avoids the blindness in selecting the gain matrix. According to the simulation analysis of the same model, the state estimation errors of the two observers can converge to zero quickly, which verifies the efficiency of the approach.