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 $3times 10^{-6}$ for the proposed CEPC with a ten-year running at 3$sigma$ (statistic) level, and better than $2times 10^{-6}$ 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.
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