In this paper, we propose the concept of (±)-discrete Dirac structures over a manifold by introducing (±)-discrete two-forms and incorporate discrete constraints via (±)-finite difference approximations. In particular, we develop (±)-discrete induced Dirac structures as discrete analogues of the induced Dirac structure on the cotangent bundle over a configuration manifold, as described by Yoshimura and Marsden [J. Geom. Phys. 57, 133–156 (2006)]. We demonstrate that (±)-discrete Lagrange–Dirac systems can be naturally formulated in conjunction with these (±)-induced Dirac structure. Furthermore, we show that the resulting equations of motion are equivalent to the discrete Lagrange–d’Alembert equations proposed in Cortés and Martínez [Nonlinearity 14, 1365–1392 (2001)] and McLachlan and Perlmutter [J. Nonlinear Sci. 16, 283–328 (2006)]. We also clarify the variational structure of the discrete Lagrange–Dirac dynamical systems within the framework of the (±)-discrete Lagrange–d’Alembert–Pontryagin principle. Finally, we validate the proposed discrete Lagrange–Dirac systems through numerical tests involving illustrative examples of nonholonomic systems.