The engine bleed air system (BAS) is one of the important systems for civil aircraft, and fault prediction of BAS is necessary to improve aircraft safety and the operator's profit. A dual-stage two-phase attention-based encoder-decoder (DSTP-ED) prediction model is proposed for BAS normal state estimation. Unlike traditional ED networks, the DSTP-ED combines spatial and temporal attention to better capture the spatiotemporal relationships to achieve higher prediction accuracy. Five data-driven algorithms, autoregressive integrated moving average (ARIMA), support vector regression (SVR), long short-term memory (LSTM), ED, and DSTP-ED, are applied to build prediction models for BAS. The comparison experiments show that the DSTP-ED model outperforms the other four data-driven models. An exponentially weighted moving average (EWMA) control chart is used as the evaluation criterion for the BAS failure warning. An empirical study based on Quick Access Recorder (QAR) data from Airbus A320 series aircraft demonstrates that the proposed method can effectively predict failures.