Let (M, H, gH ; g) be a sub-Riemannian manifold and (N, h) be a Riemannian manifold. For a smooth map u : M → N , we consider the energy functionaldVM , where duH is the horizontal differential of u, G : N → R is a smooth function on N . The critical maps of EG(u) are referred to as subelliptic harmonic maps with potential G. In this paper, we investigate the existence problem for subelliptic harmonic maps with potentials by a subelliptic heat flow. Assuming that the target Riemannian manifold has non-positive sectional curvature and the potential G satisfies various suitable conditions, we prove some Eells-Sampson type existence results when the source manifold is either a step-2 sub-Riemannian manifold or a step-r sub-Riemannian manifold whose sub-Riemannian structure comes from a tense Riemannian foliation.