With the recent advancement of novel biomedical technologies such as high-throughput sequencing and wearable devices, multi-modal biomedical data ranging from multi-omics molecular data to real-time continuous bio-signals are generated at an unprecedented speed and scale every day. For the first time, these multi-modal biomedical data are able to make precision medicine close to a reality. However, due to data volume and the complexity, making good use of these multi-modal biomedical data requires major effort. Researchers and clinicians are actively developing artificial intelligence (AI) approaches for data-driven knowledge discovery and causal inference using a variety of biomedical data modalities. These AI-based approaches have demonstrated promising results in various biomedical and healthcare applications. In this review paper, we summarize the state-of-the-art AI models for integrating multi-omics data and electronic health records (EHRs) for precision medicine. We discuss the challenges and opportunities in integrating multi-omics data with EHRs and future directions. We hope this review can inspire future research and developing in integrating multi-omics data with EHRs for precision medicine.