  We develop a comprehensive study of the gamma-ray flux observed by H.E.S.S.
in 5 regions of the Galactic Center (GC). Motivated by previous works on a
possible Dark Matter (DM) explanation for the TeV cut-off observed in the
innermost $\sim 16$ pc of the Galaxy, we aim to constrain the DM density
profile up to a radius $\sim 450$ pc from the GC. In this region, cosmological
simulations and Galactic dynamics studies fail to produce a strong prediction
of the DM profile. With our proof-of-concept analysis, we set upper limits on
the density distribution of thermal multi-TeV WIMPs, compatible with the
observed gamma-ray flux. The results agree with the hypothesis of a DM density
enhancement in the GC with respect to the benchmark NFW profile ($\gamma=1$)
and allow us to exclude profiles with a slope $\gamma \gtrsim 1.3$. We also
investigate the possibility that such an enhancement could be related to the
existence of a DM spike associated with the supermassive black hole Sgr A*. We
find out that the existence of an adiabatic DM spike smoothed by the scattering
off of WIMPs by the bulge stars may be consistent with the observed gamma-ray
flux if the spike forms on an underlying generalized NFW profile with $\gamma
\lesssim 0.8$, corresponding to a spike slope $\gamma_{sp-star}= 1.5$ and
radius $R_\text{sp-stars} \sim 25$-$30$ pc. Instead, in the extreme case of the
instantaneous growth of the black hole, the profile could have up to $\gamma
\sim 1.2$, a corresponding $\gamma_{sp-inst}=1.4$ and $R_\text{sp-inst}\sim
15$-$25$ pc. Moreover, the results of our analysis of the total DM mass
enclosed within the S2 orbit are less stringent than the spectral analysis. Our
work aims to guide future studies of the GC region, with both current and
next-generation telescopes, like the next Cherenkov Telescope Array, that will
be able to scan the GC with improved flux sensitivity and angular resolution.
