It is the aim of this paper to prove that for an arbitrary metric space (X, d) and a set Δ of nonempty closed subsets of X which contains all singletons and which is closed under enlargements, we can construct a canonical approach distance on the hyperspace CL(X), having the Δ-proximal topology (resp. the Hausdorff metric) as its topological (resp. ∞ p-metric) coreflection. We investigate some properties like, e.g., compactness and completeness of the introduced approach structures. In this way we obtain results which generalize their classical counterparts for proximal ‘hit-and-miss’ hypertopologies. We also give a characterization of the completion of the introduced approach spaces.