Gaze-contingent displays have been widely used in vision research and virtual reality applications. Due to data transmission, image processing, and display preparation, the time delay between the eye tracker and the monitor update may lead to a misalignment between the eye position and the image manipulation during eye movements. We propose a method to reduce the misalignment using a Taylor series to predict the saccadic eye movement. The proposed method was evaluated using two large datasets including 219,335 human saccades (collected with an EyeLink 1000 system, 95% range from 1° to 32°) and 21,844 monkey saccades (collected with a scleral search coil, 95% range from 1° to 9°). When assuming a 10-ms time delay, the prediction of saccade movements using the proposed method could reduce the misalignment greater than the state-of-the-art methods. The average error was about 0.93° for human saccades and 0.26° for monkey saccades. Our results suggest that this proposed saccade prediction method will create more accurate gaze-contingent displays.