Context. The future Euclid space satellite mission will offer an invaluable opportunity to constrain modifications to Einstein's general relativity at cosmic scales. In this paper, we focus on modified gravity models characterised, at linear scales, by a scale-independent growth of perturbations while featuring different testable types of derivative screening mechanisms at smaller non-linear scales. Aims. We considered three specific models, namely Jordan-Brans-Dicke, a scalar-tensor theory with a flat potential, the normal branch of Dvali-Gabadadze-Porrati (nDGP) gravity, a braneworld model in which our Universe is a four-dimensional brane embedded in a five-dimensional Minkowski space-time, and k-mouflage gravity, an extension of k-essence scenarios with a universal coupling of the scalar field to matter. In preparation for real data, we provide forecasts from spectroscopic and photometric primary probes by Euclid on the cosmological parameters and the additional parameters of the models, respectively, ωBD, Ωgc and ϵ2,0, which quantify the deviations from general relativity. This analysis will improve our knowledge of the cosmology of these modified gravity models. Methods. The forecast analysis employs the Fisher matrix method applied to weak lensing (WL); photometric galaxy clustering (GCph), spectroscopic galaxy clustering (GCsp) and the cross-correlation (XC) between GCph and WL. For the Euclid survey specifications, we define three scenarios that are characterised by different cuts in the maximum multipole and wave number, to assess the constraining power of non-linear scales. For each model we considered two fiducial values for the corresponding model parameter. Results. In an optimistic setting at 68.3% confidence interval, we find the following percentage relative errors with Euclid alone: for log10 ωBD, with a fiducial value of ωBD = 800, 27.1% using GCsp alone, 3.6% using GCph+WL+XC and 3.2% using GCph+WL+XC+GCsp; for log10 Ωgc, with a fiducial value of Ωgc = 0.25, we find 93.4, 20 and 15% respectively; and finally, for ϵ2,0 = ‑0.04, we find 3.4%, 0.15%, and 0.14%. From the relative errors for fiducial values closer to their ΛCDM limits, we find that most of the constraining power is lost. Our results highlight the importance of the constraining power from non-linear scales. ★This paper is published on behalf of the Euclid Consortium.