alpha -Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to alpha -harmonic maps that were introduced by Sacks–Uhlenbeck to attack the existence problem for harmonic maps from closed surfaces. For alpha >1, the latter are known to satisfy a Palais–Smale condition, and so, the technique of Sacks–Uhlenbeck consists in constructing alpha -harmonic maps for alpha >1 and then letting alpha rightarrow 1. The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed alpha -Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth if the perturbation function is smooth. By varepsilon -regularity and suitable perturbations, we can then show that such a sequence of perturbed alpha -Dirac-harmonic maps converges to a smooth coupled alpha -Dirac-harmonic map.