The effect of a Kerr nonlinearity on dynamical tunneling is studied, using coupled whispering gallery modes in an optical microcavity. The model system that we have chosen is the “add-drop filter,” which is comprised of an optical microcavity and two waveguides coupled to the cavity. Due to the evanescent fields scattering on the waveguide, the whispering gallery modes in the microcavity form doublets, which manifest themselves as splittings in the spectrum. As these doublets can be regarded as a spectral feature of dynamical tunneling between two different dynamical states with a spatial overlap, the effect of a Kerr nonlinearity on the doublets is numerically investigated in the more general context of the relationship between cubic nonlinearity and dynamical tunneling. Within the numerical realization of the model system, it is observed that the doublets show a bistable transition in their transmission curve as the Kerr nonlinearity in the cavity is increased. At the same time, one rotational mode becomes dominant over the other one in the transmission, since the two states in the doublet have uneven linewidths. By using coupled-mode theory, the underlying mode dynamics of the phenomena is theoretically modeled and clarified.