In this work, we construct an interacting model of the Rényi holographic dark energy in the Brans-Dicke theory of gravity using Rényi entropy in a spatially flat Friedmann-Lemaître-Robertson-Walker Universe considering the infrared cut-off as the Hubble horizon. In this setup, we then study the evolutionary history of some important cosmological parameters, in particular, deceleration parameter, Hubble parameter, equation of state parameter, and Rényi holographic dark energy density parameter in both nonflat Universe and flat Universe scenarios and also observe satisfactory behaviors of these parameters in the model. We find that during the evolution, the present model can give rise to a late-time accelerated expansion phase for the Universe preceded by a decelerated expansion phase for both flat and nonflat cases. Moreover, we obtain as , which indicates that this model behaves like the cosmological constant at the future. The stability analysis for the distinct estimations of the Rényi parameter and coupling coefficient has been analyzed. The results indicate that the model is stable at the late time.