With the goal of investigating the nature of O vi absorbers at high redshifts, we study the effects of proximity to the background quasar. In a sample of 16 quasars at zQSO between 2.14 and 2.87 observed at high signal‐to‐noise ratio and 6.6 km s−1 resolution with the Very Large Telescope/Ultraviolet and Visual Echelle Spectrograph (VLT/UVES), we detect 35 O vi absorption‐line systems (comprising over 100 individual O vi components) lying within 8000 km s−1 of zQSO. We present component fits to the O vi absorption and the accompanying H i, C iv and N v. The systems can be categorized into nine strong and 26 weak O vi absorbers. The strong (intrinsic) absorbers are defined by the presence of either broad, fully saturated O vi absorption or partial coverage of the continuum source, and in practice all have log N(O vi) ≳ 15.0; these systems are interpreted as representing either quasar‐driven outflows or gas close to the central engine of the active galactic nuclei. The weak (also known as narrow) systems show no partial coverage or saturation, and are characterized by log N(O vi) < 14.5 and a median total velocity width of only 42 km s−1. The incidence dN/dz of weak O vi systems within 2000 km s−1 of the quasar down to a limiting equivalent width of 8 mÅ is 42 ± 12. Between 2000 and 8000 km s−1, dN/dz falls to 14 ± 4, equal to the incidence of intervening O vi absorbers measured in the same spectra. Whereas the accompanying H i and C iv column densities are significantly lower (by a mean of ∼1 dex) in the weak O vi absorbers within 2000 km s−1 of zQSO than in those at larger velocities, the O vi column densities display no dependence on proximity. Furthermore, significant offsets between the H i and O vi centroids in ≈50 per cent of the weak absorbers imply that (at least in these cases) the H i and O vi lines are not formed in the same phase of gas, preventing us from making reliable metallicity determinations and ruling out single‐phase photoionization model solutions. In summary, we find no firm evidence that quasar radiation influences the weak O vi absorbers, suggesting they are collisionally ionized rather than photoionized, possibly in the multiphase haloes of foreground galaxies. Non‐equilibrium collisional ionization models are needed to explain the low temperatures in the absorbing gas, which are implied by narrow‐line widths (b < 14 km s−1) in over half of the observed O vi components.