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Neural processing of sensory input in the brain takes time, and for that reason our awareness of visual events lags behind their actual occurrence. One way the brain might compensate to minimize the impact of the resulting delays is through extrapolation. Extrapolation mechanisms have been argued to underlie perceptual illusions in which moving and static stimuli are mislocalised relative to one another (such as the flash-lag and related effects). However, where in the visual hierarchy such extrapolation processes take place remains unknown. Here, we address this question by identifying monocular and binocular contributions to the flash-grab illusion. In this illusion, a brief target is flashed on a moving background that reverses direction. As a result, the perceived position of the target is shifted in the direction of the reversal. We show that the illusion is attenuated, but not eliminated, when the motion reversal and the target are presented dichoptically to separate eyes. This reveals extrapolation mechanisms at both monocular and binocular processing stages contribute to the illusion. We interpret the results in a hierarchical predictive coding framework, and argue that prediction errors in this framework manifest directly as perceptual illusions.