  In the paper, we make a comprehensive study on the hadroproduction of the
$B_c (B_c^*)$ meson via the gluon-gluon fusion mechanism at the RHIC and LHC
colliders. Total and differential cross sections via the proton-nucleus ($p$-N)
and nucleus-nucleus (N-N) collision modes have been discussed under various
collision energies. To compare with those via the proton-proton collision mode
at the LHC, we observe that sizable number of $B_c (B_c^*)$-meson events can
also be produced via the $p$-N and N-N collision modes at the RHIC and LHC. If
assuming the spin-triplet $B^*_c$ meson directly decays to the spin-singlet
$B_c$ meson with $100\%$ probability, $1.2 \times 10^5$ and $4.7 \times 10^5$
$B_c$-meson events can be produced via the $p$-Au and Au-Au collision modes at
the RHIC in one operation year; $5.8 \times 10^6$ and $4.6 \times 10^6$
$B_c$-meson events can be produced via the $p$-Pb and Pb-Pb collision modes at
the LHC in one operation year.
