The production of quantum entanglement between weakly coupled mapping systems, whose classical counterparts are both strongly chaotic, is investigated. In the weak-coupling regime, it is shown that time-correlation functions of the unperturbed systems determine the entanglement production. In particular, we elucidate that the increment of the nonlinear parameter of coupled kicked tops does not accelerate the entanglement production in the strongly chaotic region. An approach to the dynamical inhibition of entanglement is suggested.