As part of the Euclid Early Release Observations (ERO) programme, we analyse deep, wide-field imaging from the VIS and NISP instruments of two Milky Way globular clusters (GCs), namely NGC 6254 (M10) and NGC 6397, to look for observational evidence of their dynamical interaction with the Milky Way. We search for such an interaction in the form of structural and morphological features in the clusters' outermost regions, which are suggestive of the development of tidal tails on scales larger than those sampled by the ERO programme. Our multi-band photometric analysis results in deep and well-behaved colour-magnitude diagrams that, in turn, enable an accurate membership selection. The surface brightness profiles built from these samples of member stars are the deepest ever obtained for these two Milky Way GCs, reaching down to $\sim30.0$ mag~arcsec$^{-2}$, which is about $1.5$ mag arcsec$^{-2}$ below the current limit. The investigation of the two-dimensional density map of NGC 6254 reveals an elongated morphology of the cluster peripheries in the direction and with the amplitude predicted by $N$-body simulations of the cluster's dynamical evolution, at high statistical significance. We interpret this as strong evidence for the first detection of tidally induced morphological distortion around this cluster. The density map of NGC 6397 reveals a slightly elliptical morphology, in agreement with previous studies, which requires further investigation on larger scales to be properly interpreted. This ERO project thus demonstrates the power of Euclid in studying the outer regions of GCs at an unprecedented level of detail, thanks to the combination of large field of view, high spatial resolution, and depth enabled by the telescope. Our results highlight the future Euclid survey as the ideal data set to investigate GC tidal tails and stellar streams.