Population biobanks are an increasingly important infrastructure to support research and will be a much-needed resource in the delivery of personalised medicine. Artificial intelligence (AI) systems can process and cross-link very large amounts of data quickly and be used not only for improving research power but also for helping with complex diagnosis and prediction of diseases based on health profiles. AI, therefore, potentially has a critical role to play in personalised medicine, and biobanks can provide a lot of the necessary baseline data related to healthy populations that will enable the development of AI tools. To develop these tools, access to personal data, and in particular, sensitive data, is required. Such data could be accessed from biobanks. Biobanks are a valuable resource for research but accessing and using the data contained within such biobanks raise a host of legal, ethical, and social issues (ELSI). This includes the appropriate consent to manage the collection, storage, use, and sharing of samples and data, and appropriate governance models that provide oversight of secondary use of samples and data. Biobanks have developed new consent models and governance tools to enable access that address some of these ELSI-related issues. In this paper, we consider whether such governance frameworks can enable access to biobank data to develop AI. As Italy has one of the most restrictive regulatory frameworks on the use of genetic data in Europe, we examine the regulatory framework in Italy. We also look at the proposed changes under the European Health Data Space (EHDS). We conclude by arguing that currently, regulatory frameworks are misaligned and unless addressed, accessing data within Italian biobanks to train AI will be severely limited.