We present measurements of the intergalactic medium (IGM) temperature within ∼5 proper Mpc of seven luminous quasars at z ≃ 6. The constraints are obtained from the Doppler widths of Lyα absorption lines in the quasar near zones and build upon our previous measurement for the z = 6.02 quasar SDSS J0818+1722. The expanded data set, combined with an improved treatment of systematic uncertainties, yields an average temperature at the mean density of () at 68 (95) per cent confidence for a flat prior distribution over 3.2 ≤ log (T0/K) ≤ 4.8. In comparison, temperatures measured from the general IGM at z ≃ 5 are ∼0.3 dex cooler, implying an additional source of heating around these quasars which is not yet present in the general IGM at slightly lower redshift. This heating is most likely due to the recent reionization of He ii in vicinity of these quasars, which have hard and non-thermal ionizing spectra. The elevated temperatures may therefore represent evidence for the earliest stages of He ii reionization in the most biased regions of the high-redshift Universe. The temperature as a function of distance from the quasars is consistent with being constant, log (T0/K) ≃ 4.2, with no evidence for a line-of-sight thermal proximity effect. However, the limited extent of the quasar near zones prevents the detection of He iii regions larger than ∼5 proper Mpc. Under the assumption that the quasars have reionized the He ii in their vicinity, we infer that the data are consistent with an average optically bright phase of duration in excess of 106.5 yr. These measurements represent the highest redshift IGM temperature constraints to date, and thus provide a valuable data set for confronting models of H i reionization.