This paper revisits a well-known Tsypkin criterion for stability analysis of discrete-time nonlinear Lur'e systems. When nonlinearities are monotonic and sector restricted by [0, <(Delta)over bar>], where <(Delta)over bar> is positive definite, it is shown by Kapila and Haddad that the system is absolutely stable if a function G(0)(z) = <(Delta)over bar>(-1) + {I + (1 - z(-1))K+}G(z) is strictly positive real, where K+ is nonnegative diagonal and G(z) represents a transfer function of the linear part of the Lur'e system with invertible or identically zero G(0). This paper extends this criterion when <(Delta)over bar> is positive diagonal, by choosing a new Lyapunov function to obtain an LMI criterion. From a frequency-domain interpretation of this LMI criterion, another sufficient criterion is generated which establishes that the system is absolutely stable if a function G(0)(z) = <(Delta)over bar>(-1) + {I + (1 - z(-1)) K+ + (1 - z)K-} G(z) is strictly positive real, where K+ and K- are nonnegative diagonal and orthogonal to each other.