Summary: Let \(f:M^n\to\mathbb R^{n+p}\) be an isometric immersion of an \(n\)-dimensional Riemannian manifold \(M^n\) into the \((n+p)\)-dimensional Euclidean space. Its Gauss map \(\phi:M^n\to G_n(\mathbb R^{n+p})\) into the Grassmannian \(G_n(\mathbb R^{n+p})\) is defined by assigning to every point of \(M^n\) its tangent space, considered as a vector subspace of \(\mathbb R^{n+p}\). The third fundamental form III of \(f\) is the pullback of the canonical Riemannian metric on \(G_p(\mathbb R^{n+p})\) via \(\phi\). In this article we derive a complete classification of all those \(f\) with codimension two for which the Gauss map \(\phi\) is homothetic; i.e., III is a constant multiple of the Riemannian metric on \(M^n\). We furthermore study and classify codimension two submanifolds with homothetic Gauss map in real space forms of nonzero curvature. To conclude, based on a strong connection established between homothetic Gauss map and minimal Einstein submanifolds, we pose a conjecture suggesting a possible complete classification of the submanifolds with the former property in arbitrary codimension.