We use state-of-the-art hydrodynamical simulations combined with a one-dimensional radiative transfer code to assess the extent to which the highly ionized regions observed close to z≃ 6 quasars, which we refer to as near-zones, can constrain the ionization state of the surrounding intergalactic medium (IGM). We find the appearance in Lyα absorption of a quasar H ii ionization front expanding into a neutral IGM can be very similar to a classical proximity zone, produced by the enhancement in ionizing flux close to a quasar embedded in a highly ionized IGM. The observed sizes of these highly ionized near-zones and their redshift evolution can be reproduced for a wide range of IGM neutral hydrogen fractions for plausible values of the luminosity and lifetime of the quasars. The observed near-zone sizes at the highest observed redshifts are equally consistent with a significantly neutral and a highly ionized surrounding IGM. Stronger constraints on the IGM neutral hydrogen fraction can be obtained by considering the relative size of the near-zones in the Lyα and Lyβ regions of a quasar spectrum. A large sample of high-quality quasar absorption spectra with accurate determinations of near-zone sizes and their redshift evolution in both the Lyα and Lyβ regions should confirm or exclude the possibility that the Universe is predominantly neutral at the highest observed redshifts. The width of the discrete absorption features in these near-zones will contain important additional information on the ionization state and the previous thermal history of the IGM at these redshifts.