  In this paper, we make a detailed study on the production of the $B_c$,
$B_c^*$, $B_c(2\,^1S_0)$, and $B^*_c(2\,^3S_1)$ mesons via the three planned
running modes ($Z$, $W$, and $H$) at the future Circular Electron-Positron
Collider (CEPC). The fragmentation-function and full nonrelativistic QCD
approaches are adopted to calculate the production cross sections. Considering
the excited states shall decay to the ground $B_c$ state (e.g. $1^1S_0$-state)
with almost $100\%$ probability, our numerical results show that up to
next-to-leading order QCD corrections, there are about $1.4\times 10^8$ $B_c$
events to be accumulated via the $Z$ mode ($\sqrt{s}=m_{_Z}$) of the CEPC, but
only about $1.6\times 10^4$ and $1.1\times 10^4$ $B_c$ events to be accumulated
via the $W$ ($\sqrt{s}=160\,{\rm GeV}$) and $H$ ($\sqrt{s}=240\,{\rm GeV}$)
modes, respectively. Since the $Z$ mode is the best mode among the three
planned modes of the CEPC for studying the production of the $B_c$, $B_c^*$,
$B_c(2\,^1S_0)$, and $B^*_c(2\,^3S_1)$ mesons and the differential
distributions of these mesons may be measured precisely at this mode, we
further present the differential cross sections $d\sigma/(dz\,d{\rm
cos}\theta)$ and $d\sigma/dz$ via the $Z$ mode of the CEPC.
