Osteoporosis is a common chronic metabolic bone disease often under-diagnosed and under-treated due to the limited access to bone mineral density (BMD) examinations, e.g., via Dual-energy X-ray Absorptiometry (DXA). This paper proposes a method to predict BMD from Chest X-ray (CXR), one of the most commonly accessible and low-cost medical imaging examinations. The proposed method first automatically detects Regions of Interest (ROIs) of local CXR bone structures. Then a multi-ROI deep model with transformer encoder is developed to exploit both local and global information in the chest X-ray image for accurate BMD estimation. The proposed method is evaluated on 13719 CXR patient cases with ground truth BMD measured by the gold standard DXA. The model predicted BMD has a strong correlation with the ground truth (Pearson correlation coefficient 0.894 on lumbar 1). When applied in osteoporosis screening, it achieves a high classification performance (average AUC of 0.968). As the first effort of using CXR scans to predict the BMD, the proposed algorithm holds strong potential to promote early osteoporosis screening and public health.