We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc D constrained by Hardy and radial-weighted Bergman norms. We find sharp asymptotics on the corresponding interpolation constants. As another application of our techniques we prove embedding theorems for rational functions. We find that the embedding of H ^∞ into Hardy or radial-weighted Bergman spaces in D is invertible on the subset of rational functions of a given degree n whose poles are separated from the unit circle and obtain asymptotically sharp estimates of the corresponding embedding constants.