TREX-DM is conceived to look for low-mass Weakly Interacting Massive Particles (WIMPs) using a gas Time Projection Chamber equipped with Micromegas readout planes at the Canfranc Underground Laboratory. The detector can hold in the active volume $$\sim 20~l$$∼20l of pressurized gas up to 10 bar, corresponding to 0.30 kg of Ar or 0.16 kg of Ne. The Micromegas are read with a self-triggered acquisition, being thresholds below 0.4 keV (electron equivalent) at reach. A low background level in the lowest energy region is another essential requirement. To assess the expected background, all the relevant sources have been considered, including the measured fluxes of gamma radiation, muons and neutrons at the Canfranc Laboratory, together with the activity of most of the components used in the detector and ancillary systems, obtained in a complete assay program. The background contributions have been simulated by means of a dedicated application based on Geant4 and a custom-made code for the detector response. The background model developed for the detector presently installed in Canfranc points to levels from 1 to 10 counts $$\hbox {keV}^{-1}\ \hbox {kg}^{-1}\ \hbox {day}^{-1}$$keV-1kg-1day-1 in the region of interest, making TREX-DM competitive in the search for low-mass WIMPs. A roadmap to further decrease it down to 0.1 counts $$\hbox {keV}^{-1}\ \hbox {kg}^{-1}\ \hbox {day}^{-1}$$keV-1kg-1day-1 is underway.