In this paper, we consider perturbations to a sequence of moments associated with an orthogonality linear functional that is represented by a positive measure supported in (−1,1). In particular, given a perturbation to such a measure on the real line, we analyze the perturbation obtained on the corresponding measure on the unit circle, when both measures are related through the Szeg´´ o transformation. A similar perturbation is analyzed through the inverse Szeg´´ o transformation. In both cases, we show that the applied perturbation can be expressed in terms of the singular part of the measures, and also in terms of the corresponding sequences of moments.