We prove the renormalizability to all orders of a refined Gribov–Zwanziger type action in linear covariant gauges in four-dimensional Euclidean space. In this model, the Gribov copies are taken into account by requiring that the Faddeev–Popov operator is positive definite with respect to the transverse component of the gauge field, a procedure which turns out to be analogous to the restriction to the Gribov region in the Landau gauge. The model studied here can be regarded as the first approximation of a more general nonperturbative BRST invariant formulation of the refined Gribov–Zwanziger action in linear covariant gauges obtained recently in Carpi et al. (2015, 0000). A key ingredient of the set up worked out in Carpi et al. (2015, 0000) is the introduction of a gauge invariant field configuration Aμ which can be expressed as an infinite non-local series in the starting gauge field Aμ. In the present case, we consider the approximation in which only the first term of the series representing Aμ is considered, corresponding to a pure transverse gauge field. The all order renormalizability of the resulting action gives thus a strong evidence of the renormalizability of the aforementioned more general nonperturbative BRST invariant formulation of the Gribov horizon in linear covariant gauges.