We present a systematic analysis of X-ray archival data of all the 29 quasars (QSOs) at z> 5.5 observed so far with Chandra, XMM-Newton and Swift-XRT, including the most-distant quasar ever discovered, ULAS J1120+0641 (z = 7.08). This study allows us to place constraints on the mean spectral properties of the primordial population of luminous Type 1 (unobscured) quasars. Eighteen quasars are detected in the X-ray band, and we provide spectral-fitting results for their X-ray properties, while for the others we provide upper limits to their soft (0.5–2.0 keV) X-ray flux. We measured the power-law photon index and derived an upper limit to the column density for the five quasars (J1306+0356, J0100+2802, J1030+0524, J1148+5251, J1120+0641) with the best spectra (>30 net counts in the 0.5–7.0 keV energy range) and find that they are consistent with values from the literature and lower-redshift quasars. By stacking the spectra of ten quasars detected by Chandra in the redshift range 5.7 ≤ z ≤ 6.1 we find a mean X-ray power-law photon index of Γ = 1.92-0.27+0.28 and a neutral intrinsic absorption column density of NH ≤ 1023 cm-2. These results suggest that the X-ray spectral properties of luminous quasars have not evolved up to z ≈ 6. We also derived the optical-X-ray spectral slopes (αox) of our sample and combined them with those of previous works, confirming that αox strongly correlates with UV monochromatic luminosity at 2500 Å. These results strengthen the non-evolutionary scenario for the spectral properties of luminous active galactic nuclei (AGN).