Abstract: Euclid will image 10^5 new strong lenses at ~100mas resolution, providing a dataset of unprecedented size with which to search for dark matter subhaloes and test CDM. However, traditional methods for subhalo detection cannot scale to a dataset of this size. We introduce a machine learning method for estimating the sensitivity of strong lens images to dark matter subhaloes in the lens. We then estimate the sensitivity in 16,000 simulated Euclid strong lens observations, to determine how useful Euclid is for constraining DM models in this way. We find that, assuming a 3 sigma detection threshold, 2.35 per cent of the area inside twice the Einstein radius is sensitive enough to detect a subhalo with M_max = 10^10 M_sun. The best pixel in the dataset is sensitive at M_max = 10^(8.8±0.2) M_sun. We find that, assuming CDM, the average Euclid lens should have 0.0143 detectable subhaloes, or one detection in every ~70 lenses. However, in the most sensitive lenses, this increases dramatically to 0.356 detectable subhaloes, or one for every ~3 lenses. Assuming Euclid discovers 170,000 strong lenses as predicted, we expcet ~2500 new subhalo detections. However, the sensitivity limit that we find means these detections cannot differentiate between CDM and WDM models, although they can place strong constraints on the normalisation of the subhalo mass function.