We present a study, based on archival XMM–Newton observations, of the extended X-ray emission associated with the inner disc of M33. After the exclusion of point sources with LX > 2 × 1035 erg s−1 (0.3–6 keV), we investigate both the morphology and spectrum of the residual X-ray emission, comprising the integrated signal from unresolved discrete sources and diffuse components. This residual emission has a soft X-ray spectrum which can be fitted with a two-temperature thermal model, with kT≈ 0.2 keV and ≈0.6 keV, the cooler component accounting for the bulk of the luminosity. There is evidence that the X-ray emitting plasma has a subsolar metal abundance. The soft X-ray surface brightness distribution shows a strong correlation with far-ultraviolet (FUV) emission and since the latter serves as a tracer of the inner spiral arms of M33 this is indicative of a close connection between recent star formation activity and the production of soft X-rays. Within 3.5 kpc of the nucleus of M33, the soft X-ray and FUV surface brightness distributions exhibit similar radial profiles. The implication is that the ratio of the soft X-ray luminosity (0.3–2.0 keV) per unit disc area to the star formation rate per unit disc area remains fairly constant within this inner disc region. We derive a value for this ratio of 1–1.5 × 1039 erg s−1 (M⊙ yr−1)−1, which is towards the top of the range of similar estimates for several other nearby face-on spiral galaxies (e.g. M51, M83). In the same region, the ratio of soft X-ray luminosity to stellar mass (the latter derived from K-band photometry) is 4 × 1028 erg s−1 M−1⊙, a factor of 5–10 higher than is typical of dwarf elliptical galaxies (e.g. M32, NGC 3379), suggesting that 10–20 per cent of the unresolved emission seen in M33 may originate in its old stellar population. The remainder of the soft X-ray emission is found to be equally split between two spatial components, one which closely traces the spiral arms of the galaxy and the other more smoothly distributed across the inner disc of M33. The former must represent a highly clumped low-filling-factor component linked to sites of recent or ongoing star formation, encompassing H ii regions, X-ray source complexes and X-ray superbubbles, whereas the distribution of latter gives few clues as to its exact origin.