  We present precise abundance determinations of two near-pristine damped
Ly$\alpha$ systems (DLAs) to assess the nature of the [O/Fe] ratio at [Fe/H] <
-3 (i.e. <1/1000 of the solar metallicity). Prior observations indicate that
the [O/Fe] ratio is consistent with a constant value, [O/Fe] ~ +0.4, when -3 <
[Fe/H] < -2, but this ratio may increase when [Fe/H] < -3. In this paper, we
test this picture by reporting new, high-precision [O/Fe] abundances in two of
the most metal-poor DLAs currently known. We derive values of [O/Fe] = +0.50
+/- 0.10 and [O/Fe] = +0.62 +/- 0.05 for these two z ~ 3 near-pristine gas
clouds. These results strengthen the idea that the [O/Fe] abundances of the
most metal-poor DLAs are elevated compared to DLAs with [Fe/H] > -3. We compare
the observed abundance pattern of the latter system to the nucleosynthetic
yields of Population III supernovae (SNe), and find that the enrichment can be
described by a (19-25) M$_{\odot}$ Population III SN that underwent a
(0.9-2.4)$\times 10^{51}$ erg explosion. These high-precision measurements
showcase the behaviour of [O/Fe] in the most metal-poor environments. Future
high-precision measurements in new systems will contribute to a firm detection
of the relationship between [O/Fe] and [Fe/H]. These data will reveal whether
we are witnessing a chemical signature of enrichment from Population III stars
and allow us to rule out contamination from Population II stars.
