It is necessary to clarify the responsibilities and competences on national and Union level as regards AI systems that are built on general-purpose AI models. To avoid overlapping competences, where an AI system is based on a general-purpose AI model and the model and system are provided by the same provider, the supervision should take place at Union level through the AI Office, which should have the powers of a market surveillance authority within the meaning of Regulation (EU) 2019/1020 for this purpose. In all other cases, national market surveillance authorities remain responsible for the supervision of AI systems. However, for general-purpose AI systems that can be used directly by deployers for at least one purpose that is classified as high-risk, market surveillance authorities should cooperate with the AI Office to carry out evaluations of compliance and inform the Board and other market surveillance authorities accordingly. Furthermore, market surveillance authorities should be able to request assistance from the AI Office where the market surveillance authority is unable to conclude an investigation on a high-risk AI system because of its inability to access certain information related to the general-purpose AI model on which the high-risk AI system is built. In such cases, the procedure regarding mutual assistance in cross-border cases in Chapter VI of Regulation (EU) 2019/1020 should apply by analogy.