We address the issue of the renormalizability of the gauge-invariant nonlocal dimension-two operator Amin2, whose minimization is defined along the gauge orbit. Despite its nonlocal character, we show that the operator Amin2 can be cast in local form through the introduction of an auxiliary Stueckelberg field. The localization procedure gives rise to an unconventional kind of Stueckelberg-type action that turns out to be renormalizable to all orders of perturbation theory. In particular, as a consequence of its gauge invariance, the anomalous dimension of the operator Amin2 turns out to be independent from the gauge parameter α entering the gauge-fixing condition, thus being given by the anomalous dimension of the operator A2 in the Landau gauge.