We review existing ROSAT detections of single Galactic Wolf-Rayet (WR) stars and develop wind models to interpret the X-ray emission. The ROSAT data, consisting of bandpass detections from the ROSAT All-Sky Survey (RASS) and some pointed observations, exhibit no correlations of the WR X-ray luminosity (LX) with any star or wind parameters of interest (e.g. bolometric luminosity, mass-loss rate or wind kinetic energy), although the dispersion in the measurements is quite large. The lack of correlation between X-ray luminosity and wind parameters among the WR stars is unlike that of their progenitors, the O stars, which show trends with such parameters. In this paper we seek to (i) test by how much the X-ray properties of the WR stars differ from the O stars and (ii) place limits on the temperature TX and filling factor fX of the X-ray-emitting gas in the WR winds. Adopting empirically derived relationships for TX and fX from O-star winds, the predicted X-ray emission from WR stars is much smaller than observed with ROSAT. Abandoning the TX relation from O stars, we maximize the cooling from a single-temperature hot gas to derive lower limits for the filling factors in WR winds. Although these filling factors are consistently found to be an order of magnitude greater than those for O stars, we find that the data are consistent (albeit the data are noisy) with a trend of in WR stars, as is also the case for O stars.