Background: The integration and analysis of multi-modal data are increasingly essential across various domains including bioinformatics. As the volume and complexity of such data grow, there is a pressing need for computational models that not only integrate diverse modalities but also leverage their complementary information to improve clustering accuracy and insights, especially when dealing with partial observations with missing data. Results: We propose Generalized Probabilistic Canonical Correlation Analysis (GPCCA), an unsupervised method for the integration and joint dimensionality reduction of multi-modal data. GPCCA addresses key challenges in multi-modal data analysis by handling missing values within the model, enabling the integration of more than two modalities, and identifying informative features while accounting for correlations within individual modalities. The model demonstrates robustness to various missing data patterns and provides low-dimensional embeddings that facilitate downstream clustering and analysis. In a range of simulation settings, GPCCA outperforms existing methods in capturing essential patterns across modalities. Additionally, we demonstrate its applicability to multi-omics data from TCGA cancer datasets and a multi-view image dataset. Conclusion: GPCCA offers a useful framework for multi-modal data integration, effectively handling missing data and providing informative low-dimensional embeddings. Its performance across cancer genomics and multi-view image data highlights its robustness and potential for broad application. To make the method accessible to the wider research community, we have released an R package, GPCCA, which is available at https://github.com/Kaversoniano/GPCCA.