  One of the interesting portals linking a dark sector and the standard model
(SM) is the kinetic mixing between the SM $U(1)_Y$ field with a new dark photon
$A'$ from a $U(1)_{A'}$ gauge interaction. Stringent limits have been obtained
for the kinetic mixing parameter $\epsilon$ through various processes. In this
work, we study the possibility of searching for a dark photon interaction at a
circular $e^+e^-$ collider through the process $e^+ e^-\to \gamma A^{\prime *}
\to \gamma \mu^+\mu^-$. We find that the constraint on $\epsilon^2$ for dark
photon mass in the few tens of GeV range, assuming that the $\mu^+\mu^-$
invariant mass can be measured to an accuracy of $0.5\%m_{A'}$, can be better
than $3\times 10^{-6}$ for the proposed CEPC with a ten-year running at
3$\sigma$ (statistic) level, and better than $2\times 10^{-6}$ for FCC-ee with
even just one-year running at $\sqrt{s} = 240$ GeV, better than the LHC and
other facilities can do in a similar dark photon mass range. For FCC-ee,
running at $\sqrt{s}=160$ GeV, the constraint can be even better.
