The utilization of Low Earth Orbit (LEO) satellites is anticipated to augment various aspects of traditional GNSS-based Positioning, Navigation, and Timing (PNT) services. While the LEO satellite orbital products can nowadays be produced with rather high accuracy in real-time of a few centimeters, the precision of the LEO satellite clock products that can be achieved in real-time is less studied. The latter, however, plays an essential role in the LEO-augmented positioning and timing performances. In real-time, the users eventually use the predicted LEO satellite clocks, with their precision determined by both the near-real-time clock precision and the prediction time needed to match the time window for real-time applications, i.e., the precision loss during the prediction phase. In this study, a real-time LEO satellite clock determination method, consisting of near-real-time clock determination with ultra-short-term clock prediction is proposed and implemented. The principles and strategies of this method are discussed in detail. The proposed method utilized Kalman-filter-based processing, but supports restarts at pre-defined times, thus hampering continuous bias propagation and accumulation from ancient epochs. Based on the method, using Sentinel-3B GNSS observations and the real-time GNSS products from the National Center for Space Studies (CNES) in France, the near-real-time LEO satellite clocks can reach a precision of 0.2 to 0.3 ns, and the precision loss during the prediction phase is within 0.07 ns for a prediction time window from 30 to 90 s. This results in a total error budget in the real-time LEO satellite clocks of about 0.3 ns.