  Black hole-main sequence star (BH-MS) binaries are one of the targets of the
future data releases of the astrometric satellite {\it Gaia}. They are supposed
to be formed in two main sites: a galactic field and star clusters. However,
previous work has never predicted the number of BH-MS binaries originating in
the latter site. In this paper, we estimate the number of BH-MS binaries formed
in open clusters and detectable with {\it Gaia} based on the results of {\it
N}-body simulations. By considering interstellar extinction in the Milky Way
(MW) and observational constraints, we predict $\sim 10$ BH-MS binaries are
observable. We also find that chemical abundance patterns of companion MSs will
help us to identify the origin of the binaries as star clusters. Such MSs are
not polluted by outflows of the BH progenitors, such as stellar winds and
supernova ejecta. Chemical anomalies might be a good test to confirm the origin
of binaries with relatively less massive MSs ($\lesssim 5M_{\odot}$), orbital
periods ($\sim 1.5\;$year) and higher eccentricities ($e \gtrsim 0.1$).
