Context.LS 5039 has been observed with several X-ray instruments so far showing quite steady emission in the long term and no signatures of accretion disk. The source also presents X-ray variability at orbital timescales in flux and photon index. The system harbors an O-type main sequence star with moderate mass-loss. At present, the link between the X-rays and the stellar wind is unclear. Aims.We study the X-ray fluxes, spectra, and absorption properties of LS 5039 at apastron and periastron passages during an epoch of enhanced stellar mass-loss, and the long term evolution of the latter in connection with the X-ray fluxes. Methods.New XMM-Newton observations were performed around periastron and apastron passages in September 2005, when the stellar wind activity was apparently higher. April 2005 Chandra observations on LS 5039 were revisited. Moreover, a compilation of H EW data obtained since 1992, from which the stellar mass-loss evolution can be approximately inferred, was carried out. Results.XMM-Newton observations show higher and harder emission around apastron than around periastron. No signatures of thermal emission or a reflection iron line indicating the presence of an accretion disk are found in the spectrum, and the hydrogen column density () is compatible with being the same in both observations and consistent with the interstellar value. 2005 Chandra observations show a hard X-ray spectrum, and possibly high fluxes, although pileup effects preclude conclusive results from being obtained. The H EW shows yearly variations of 10%, and does not seem to be correlated with X-ray fluxes obtained at similar phases, unlike what is expected in the wind accretion scenario. Conclusions.2005 XMM-Newton and Chandra observations are consistent with 2003 RXTE/PCA results, namely moderate flux and spectral variability at different orbital phases. The constancy of the seems to imply that either the X-ray emitter is located at 1012 cm from the compact object, or the density in the system is 3 to 27 times smaller than that predicted by a spherical symmetric wind model. We suggest that the multiwavelength non-thermal emission of LS 5039 is related to the observed extended radio jets and is unlikely to be produced inside the binary system.