It remains unclear whether and, if so, how nonhuman animals make on-the-fly predictions during pursuit. Here we used a novel laboratory pursuit task that incentivizes the prediction of future prey positions. We trained three macaques to perform a joystick-controlled pursuit task in which prey follow intelligent escape algorithms. Subjects aimed toward the likely future positions of the prey, which indicated that they generate internal predictions and use these to guide behavior. We then developed a generative model that explains real-time pursuit trajectories and showed that our subjects use prey position, velocity and acceleration to make predictions. We identified neurons in the dorsal anterior cingulate cortex whose responses track these three variables. These neurons multiplexed prediction-related variables with a distinct and explicit representation of the future position of the prey. Our results provide a clear demonstration that the brain can explicitly represent future predictions and highlight the critical role of anterior cingulate cortex for future-oriented cognition.