The design of the European Union (EU) DEMO reactor magnet system, currently ongoing within the EUROfusion consortium, will take advantage of the know-how developed during the design and manufacturing of ITER magnets; however, DEMO will suffer some new, more severe challenges, e.g., larger tritium inventory and higher neutron fluence, both having an impact on safety functions accomplished, among the other systems, also by the magnets. For these reasons, and in view of the need to demonstrate a high availability of the reactor (aimed at electricity production), a new, more systematic assessment of the system safety is required. As a contribution in this direction, the initiating events (IEs) of the most critical accident sequences in the EU DEMO magnet system (with special reference to the toroidal field magnets) are identified here, adopting first a functional analysis and then a failure mode, effects, and criticality analysis. In particular, the following are provided: (1) the EU DEMO magnet system is subdivided into functionally independent subsystems and components (e.g., the magnets, their cooling circuits, and their power supply system); (2) the relevant failure modes of each subsystem are systematically identified, together with the corresponding causes and consequences; (3) a list of IEs is compiled, leading to scenarios that may compromise the magnet safety and availability. Finally, the so-called postulated IEs are selected as the most challenging IEs for the safety of the magnet system. This analysis initializes a path leading to a risk-informed design, i.e., the identification of safety issues that could be addressed at the design level instead of introducing expensive mitigation measures after the design completion.