This paper investigates the dynamics of a glucose-insulin regulatory system model that incorporates: (1) insulin-degrading enzyme in the insulin equation; and (2) discrete time delays respectively in the insulin production term, hepatic glucose production term, and the insulin-degrading enzyme. We provide rigorous results of our model including the asymptotic stability of the equilibrium solution and the existence of Hopf bifurcation. We show that analytically and numerically at a certain value the time delays driven stability or instability occurs when the corresponding model has an interior equilibrium. Moreover, we illustrate the oscillatory regulation and insulin secretion via numerical simulations, which show that the model dynamics exhibit physiological observations and more information by allowing parameters to vary. Our results may provide useful biological insights into diabetes for the glucose-insulin regulatory system model.