We address the issue of the renormalizability of the gauge-invariant non-local dimension-two operator $A^2_{rm min}$, whose minimization is defined along the gauge orbit. Despite its non-local character, we show that the operator $A^2_{rm min}$ 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 which 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 $A^2_{rm min}$ turns out to be independent from the gauge parameter $alpha$ entering the gauge-fixing condition, being thus given by the anomalous dimension of the operator $A^2$ in the Landau gauge.
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