Satellite clock error plays an important role in precision positioning, so it is essential to monitor its anomalies in real time. On the premise of modeling satellite clock error, this paper proposes a real-time satellite clock anomaly monitoring algorithm, based on the forgetting factor least square algorithm (RFFLS algorithm). The principles of this algorithm are explained. Using the IGS final precision product, comparisons are made between the RFFLS algorithm, LS algorithm, and FFLS algorithm for satellite clock error prediction precision, time consumption, and anomaly monitoring performance. The results show that the time consumption of RFFLS algorithm is only one tenth of that of LS algorithm or FFLS algorithm, and the anomaly monitoring ability of RFFLS algorithm is the best. In conclusion, the RFFLS algorithm is simple, flexible, and shows a clear advantage in real-time application.