Border ownership (BO) indicates which side of a contour owns a border, and it plays a fundamental role in figure-ground segregation. The majority of neurons in V2 and V4 areas of monkeys exhibit BO selectivity. A physiological work reported that the responses of BO-selective cells show a rapid transition when a presented square is flipped along its classical receptive field (CRF) so that the opposite BO is presented, whereas the transition is significantly slower when a square with a clear BO is replaced by an ambiguous edge, e.g., when the square is enlarged greatly. The rapid transition seemed to reflect the influence of feedforward processing on BO selectivity. Herein, we investigated the role of feedforward signals and cortical interactions for time-courses in BO-selective cells by modeling a visual cortical network comprising V1, V2, and posterior parietal (PP) modules. In our computational model, the recurrent pathways among these modules gradually established the visual progress and the BO assignments. Feedforward inputs mainly determined the activities of these modules. Surrounding suppression/facilitation of early-level areas modulates the activities of V2 cells to provide BO signals. Weak feedback signals from the PP module enhanced the contrast gain extracted in V1, which underlies the attentional modulation of BO signals. Model simulations exhibited time-courses depending on the BO ambiguity, which were caused by the integration delay of V1 and V2 cells and the local inhibition therein given the difference in input stimulus. However, our model did not fully explain the characteristics of crucially slow transition: the responses of BO-selective physiological cells indicated the persistent activation several times longer than that of our model after the replacement with the ambiguous edge. Furthermore, the time-course of BO-selective model cells replicated the attentional modulation of response time in human psychophysical experiments. These attentional modulations for time-courses were induced by selective enhancement of early-level features due to interactions between V1 and PP. Our proposed model suggests fundamental roles of surrounding suppression/facilitation based on feedforward inputs as well as the interactions between early and parietal visual areas with respect to the ambiguity dependence of the neural dynamics in intermediate-level vision.