  In the present paper, we study the photoproduction of various $S$-wave $B_c$
states, i.e. $B_c(1{}^1S_0)$, $B_c^*(1{}^1S_0)$, $B_c(2{}^1S_0)$ and
$B^*_c(2{}^3S_1)$, at future $e^+e^-$ colliders within the framework of
non-relativistic QCD. Two sources of the initial photons can be realized in the
production, i.e., the LBS photon and the WWA photon. In addition to the direct
photoproduction via the sub-process, $\gamma + \gamma \rightarrow B_c + b +
\bar{c}$, we also calculate the resolved photoproduction via the sub-processes,
$\gamma + g \rightarrow B_c + b + \bar{c}$ and $q + \bar{q} \rightarrow B_c + b
+ \bar{c}$ with $q=u,d,s,g$, respectively. Numerical results indicate that the
contributions from the single resolved photoproduction are significant and even
dominant at certain collision configuration. At the future high-energy and
high-luminosity $e^+e^-$ colliders, the $B_c$ meson generated via the
photoproduction mechanism is promisingly observable and can be well studied.
