  Neutral triple gauge couplings (nTGCs) are absent in the standard model
effective theory up to dimension-6 operators, but could arise from dimension-8
effective operators. In this work, we study the pure gauge operators of
dimension-8 that contribute to nTGCs and are independent of the dimension-8
operator involving the Higgs doublet. We show that the pure gauge operators
generate both $Z\gamma Z^*$ and $Z\gamma\gamma^*$ vertices with rapid energy
dependence $\propto E^5$, which can be probed sensitively via the reaction
$e^+e^- \to Z\gamma$. We demonstrate that measuring the nTGCs via the reaction
$e^+e^- \to Z\gamma$ followed by $Z \to q\bar{q}$ decays can probe the new
physics scales of dimension-8 pure gauge operators up to the range $(1-5)$TeV
at the CEPC, FCC-ee and ILC colliders with $\sqrt{s}=(0.25-1)$TeV, and up to
the range $(10-16)$TeV at CLIC with $\sqrt{s}=(3-5)$TeV, assuming in each case
an integrated luminosity of 5/ab. We compare these sensitivities with the
corresponding probes of the dimension-8 nTGC operators involving Higgs doublets
and the dimension-8 fermionic contact operators that contribute to the
$e^+e^-Z\gamma$ vertex.
