In the Global Navigation Satellite System (GNSS), the satellite clock bias (SCB) plays an important role in the application of real-time precise point positioning (RT-PPP). Based on the operation of Beidou satellite global service, it is very important to establish a reliable Beidou SCB prediction model. In this research, an attention mechanism-based long short-term memory neural network (AttLSTM) model is applied to SCB prediction. The attention mechanism introduced in modelling can make the model pay less attention to useless information through weight allocation. In this paper, the BeiDou-3 Navigation Satellite System (BDS-3) satellite precision clock product provided by GFZ is used for clock prediction experiments. The proposed AttLSTM model, long short-term memory neural network (LSTM) model and quadratic polynomial (QP) model are compared and evaluated, and 12h and 24h SCB prediction experiments of BDS-3 satellite are set up. The results show that AttLSTM model can achieve high SCB prediction accuracy, and the averaged prediction accuracy of 12h and 24h can reach 1.41ns and 1.75ns. Compared with LSTM and QP models, the prediction accuracy of AttLSTM model is improved by 26.1%, 38.4% for 12h and 29.1%, 43.1% for 24h, respectively. Then, the clock bias predicted by the three models is applied to the static PPP positioning experiment, respectively. Through the analysis of the positioning results of 15 MGEX stations, the averaged positioning accuracy of AttLSTM model in the East, North and Up directions can reach 0.074m, 0.019m and 0.154m, respectively. Compared with LSTM and QP models, the positioning accuracy of AttLSTM model is improved by 42.5% and 44.4% in the East direction, 44.7% and 58.9% in the North direction, and 21.7% and 21.8% in the Up direction.