In this paper, we propose a discrete model reduction method that preserves the bounded realness. We define the discrete bounded real (BR) balancing and obtain the reduced order discrete models from the generalized singular perturbation approximation (GSPA) and direct truncation (DT) of the BR-balanced system. We show the properties of the reduced order systems and derive the H-x norm error bounds for the cases of the GSPA at +/-1 of the complex plane and the DT.