We have compiled a catalog of 903 quasar candidates (including known quasars) at 3 < z < 5.5 selected among X-ray sources from the XMM-Newton serendipitous survey (3XMM-DR4 catalog). We used photometric SDSS, 2MASS, and WISE data to select the objects. The surface number density of objects in our sample exceeds that in the SDSS spectroscopic quasar sample at the same redshifts by a factor of 1.5. We have performed spectroscopic observations of a subsample of new quasar candidates using a new low-and medium-resolution spectrograph at the 1.6-m AZT-33IK telescope (Mondy, Russia) and demonstrated that the purity of these candidates is about 65%. We have discovered one of the most distant (z = 5.08) X-ray selected quasars.