The Euclid mission will provide first-of-its-kind coverage in the near-infrared over deep (three fields, ∼10-20 square degrees each) and wide (∼10 000 square degrees) fields. While the survey is not designed to discover transients, the deep fields will have repeated observations over a two-week span, followed by a gap of roughly six months. In this analysis, we explore how useful the deep field observations will be for measuring properties of Type Ia supernovae (SNe Ia). Using simulations that include Euclid's planned depth, area, and cadence in the deep fields, we calculate that more than 3700 SNe between 0.0 < < 1.5 will have at least five Euclid detections around peak with signal-to-noise ratio larger than 3. While on their own, Euclid light curves are not good enough to directly constrain distances, when combined with legacy survey of space and time (LSST) deep field observations, we find that uncertainties on SN distances are reduced by 20-30 per cent for < 0.8 and by 40-50 per cent for > 0.8. Furthermore, we predict how well additional Euclid mock data can be used to constrain a key systematic in SN Ia studies - the size of the luminosity 'step' found between SNe hosted in high-mass (>1010 M⊙) and low-mass (<1010 M⊙) galaxies. This measurement has unique information in the rest-frame near-infrared (NIR). We predict that if the step is caused by dust, we will be able to measure its reduction in the NIR compared to optical at the 4σ level. We highlight that the LSST and Euclid observing strategies used in this work are still provisional and some level of joint processing is required. Still, these first results are promising, and assuming that Euclid begins observations well before the Nancy Roman Space Telescope (Roman), we expect this data set to be extremely helpful for preparation for Roman itself.