The discovery of quasars with redshifts higher than six has prompted a great deal of discussion in the literature regarding the role of quasars, both as sources of reionization, and as probes of the ionization state of the intergalactic medium (IGM). However the extreme ultraviolet (EUV) spectral index cannot be measured directly for high-redshift quasars owing to absorption at frequencies above the Lyman limit, and as a result, studies of the impact of quasars on the IGM during reionization must assume a spectral energy distribution in the EUV based on observations at low redshift, z≲ 1. In this paper we use regions of high Lyα transmission (near-zones) around the highest redshift quasars to measure the quasar EUV spectral index at z∼ 6. We jointly fit the available observations for variation of near-zone size with both redshift and luminosity, and propose that the observed relation provides evidence for an EUV spectral index that varies with absolute magnitude in the high-redshift quasar sample, becoming softer at higher luminosity. Using a large suite of detailed numerical simulations, we find that the typical value of spectral index for a luminous quasar (M1450=−27) at z∼ 6 is constrained to be α= 1.3+0.4−0.3 for a specific luminosity of the form Lν∝ν−α. This value is sensitive to the background ionization rate at z∼ 6, and we present an updated estimate of Γ= (0.18+0.18−0.09) × 10−12 s−1 as part of our analysis. We find the scatter in spectral index among individual quasars to be in the range Δα∼ 0.75–1.25. These values are in agreement with direct observations at low redshift, and indicate that there has been no significant evolution in the EUV spectral index of quasars over 90 per cent of cosmic time.