The contact values g(ij)(sigma(ij)) of the radial distribution functions of a d-dimensional mixture of (additive) hard spheres are considered. A "universality" assumption is put forward, according to which g(ij)(sigma(ij))=G(eta,z(ij)), where G is a common function for all the mixtures of the same dimensionality, regardless of the number of components, eta is the packing fraction of the mixture, and z(ij)=(sigma(i)sigma(j)/sigma(ij))[sigma(d-1)]/[sigma(d)] is a dimensionless parameter, [sigma(n)] being the nth moment of the diameter distribution. For d=3, this universality assumption holds for the contact values of the Percus-Yevick approximation, the scaled particle theory, and, consequently, the Boublik-Grundke-Henderson-Lee-Levesque approximation. Known exact consistency conditions are used to express G(eta,0), G(eta,1), and G(eta,2) in terms of the radial distribution at contact of the one-component system. Two specific proposals consistent with the above-mentioned conditions (a quadratic form and a rational form) are made for the z dependence of G(eta,z). For one-dimensional systems, the proposals for the contact values reduce to the exact result. Good agreement between the predictions of the proposals and available numerical results is found for d=2, 3, 4, and 5.