Osteoporosis is a common chronic metabolic bone disease that is 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). In this paper, we propose a method to predict BMD from Chest X-ray (CXR), one of the most commonly accessible and low-cost medical imaging examinations. Our method first automatically detects Regions of Interest (ROIs) of local and global bone structures from the CXR. 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. Our method is evaluated on 13719 CXR patient cases with their ground truth BMD scores measured by gold-standard DXA. The model predicted BMD has a strong correlation with the ground truth (Pearson correlation coefficient 0.889 on lumbar 1). When applied for osteoporosis screening, it achieves a high classification performance (AUC 0.963 on lumbar 1). As the first effort in the field using CXR scans to predict the BMD, the proposed algorithm holds strong potential in early osteoporosis screening and public health promotion.