  In this paper, we investigate the effects of a dark matter (DM) spike on the
neighborhood of Sgr A*, the black hole (BH) in the center of the Milky Way
galaxy. Our main goal is to investigate whether current and future astronomical
observations of Sgr A* could detect the presence of such a DM spike. At first,
we construct the spacetime metric around a static and spherically symmetric BH
with a DM spike, and later this solution is generalized for a rotating BH using
the Newman-Janis-Azreg A\"{i}nou algorithm. For the static BH metric, we use
the data of the S2 star orbiting the Sgr A* to determine and analyze the
constraints on the two free parameters characterizing the density and the
innermost boundary of the DM halo surrounding the BH. Furthermore, by making
use of the available observational data for the DM spike density
$\rho_\text{sp}$ and the DM spike radius $R_\text{sp}$ in the Milky Way galaxy,
we consider a geometrically-thick accretion disk model around the Sgr A* BH and
demonstrate that the effect of DM distribution on the shadow radius and the
image of the BH is considerably weak for realistic DM densities, becoming
significant only when the DM density is of the order $\rho_\text{sp} \sim
(10^{-19}-10^{-20})$ g/cm$^3$ near the BH. We further analyze the possibility
of observing this effect with radio interferometry, simulating observations
with an EHT--like array, and find that it is unlikely to be detectable in the
near future.
