We present a numerical investigation of the tunneling dynamics of a particle moving in a bistable potential with fluctuating barrier which is coupled to a non-integrable classical system and study the interplay between classical chaos and barrier fluctuation in the tunneling dynamics. We found that the coupling of the quantum system with the classical subsystem decreases the tunneling rate irrespective of whether the classical subsystem is regular or chaotic and also irrespective of the fact that whether the barrier fluctuates or not. Presence of classical chaos always enhances the tunneling rate constant. The effect of barrier fluctuation on the tunneling rate in a mixed quantum-classical system is to suppress the tunneling rate. In contrast to the case of regular subsystem, the suppression arising due to barrier fluctuation is more visible when the subsystem is chaotic.