We present the pre-concept design of the European DEMO Magnet System, which has successfully passed the DEMO plant-level gate review in 2020. The main design input parameters originate from the so-called DEMO 2018 baseline, which was produced using the PROCESS systems code. It defines a major and minor radius of 9.1 m and 2.9 m, respectively, an on-axis magnetic field of 5.3 T resulting in a peak field on the toroidal field (TF) conductor of 12.0 T. Four variants, all based on low-temperature superconductors (LTS), have been designed for the 16 TF coils. Two of these concepts were selected to be further pursued during the Concept Design Phase (CDP): the first having many similarities to the ITER TF coil concept and the second being the most innovative one, based on react-and-wind (RW) Nb3Sn technology and winding the coils in layers. Two variants for the five Central Solenoid (CS) modules have been investigated: an LTS-only concept resembling to the ITER CS and a hybrid configuration, in which the innermost layers are made of high-temperature superconductors (HTS), which allows either to increase the magnetic flux or to reduce the outer radius of the CS coil. Issues related to fatigue lifetime which emerged in mechanical analyses will be addressed further in the CDP. Both variants proposed for the six poloidal field coils present a lower level of risk for future development. All magnet and conductor design studies included thermal-hydraulic and mechanical analyses, and were accompanied by experimental tests on both LTS and HTS prototype samples (i.e. DC and AC measurements, stability tests, quench evolution etc.). In addition, magnet structures and auxiliary systems, e.g. cryogenics and feeders, were designed at pre-concept level. Important lessons learnt during this first phase of the project were fed into the planning of the CDP. Key aspects to be addressed concern the demonstration and validation of critical technologies (e.g. industrial manufacturing of RW Nb3Sn and HTS long conductors, insulation of penetrations and joints), as well as the detailed design of the overall Magnet System and mechanical structures.