Continuous blood pressure monitoring (CBPM) is critical to support the accurate prevention and reliable treatment of cardiovascular diseases. To achieve efficient multi-information interaction and further improve the monitoring performance, this research proposes an intelligent model based on transformer encoders and stacked attention gated recurrent units (TE-SAGRU) for CBPM. Long-term multi-source feature sequences with rich information are initially extracted from photoplethysmography (PPG) and electrocardiography (ECG) signals. The paralleled transformer encoders are constructed for different source feature sequences to obtain high-level feature representations and preserve respective long-term independence. The multiple stacked attention gated recurrent units are cross-connected for multi-interactive feature fusion and promoting complementarity effects of multi-source features on CBPM. Comprehensive comparison experiments are carried out to validate the effectiveness of the TE-SAGRU model, using the dataset with 1000 subjects derived from the MIMIC-III database. The continuous monitoring errors of the TE-SAGRU model for systolic blood pressure (SBP) and diastolic blood pressure (DBP) are 3.91 ± 5.65 mmHg and 2.29 ± 3.01 mmHg. The monitored results pass the requirement of the Association for the Advancement of Medical Instrumentation (AAMI) standard and achieve Grade A of the British Hypertension Society (BHS) protocol.