Abstract We develop a comprehensive study of the gamma-ray flux observed by the High Energy Stereoscopic System (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 by H.E.S.S. in the innermost ∼ 15 pc of the Galaxy, we aim to constrain the DM distribution up to a radius of ∼ 450 pc from the GC. In this region, the benchmark approach (e.g. cosmological simulations and Galactic dynamics studies) fails to produce a strong prediction of the DM profile. Within our proof-of-concept analysis, we use DRAGON to model the diffuse background emission and determine upper limits on the density distribution of thermal multi-TeV Weakly Interactive Massive Particles (WIMPs), compatible with the observed gamma-ray flux. The results are in agreement with the hypothesis of an enhancement of the DM density in the GC with respect to the benchmark Navarro-Frenk-White (NFW) profile (γ = 1) and allow us to exclude profiles with an inner slope cuspier than γ ≳ 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* at the GC. 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 γ ≲ 0.8, corresponding to a spike slope of γsp-star = 1.5 and spike radius of R sp-stars ∼ 25 30 pc. Instead, in the extreme case of the instantaneous growth of the black hole, the underlying profile could have up to γ ∼ 1.2, a corresponding γsp-inst = 1.4 and R sp-inst ∼ 15–25 pc. Finally, the results of our analysis of the total DM mass enclosed within the S2 orbit (updated with new GRAVITY data) are less stringent than the spectral analysis. Our work aims to guide future studies of the GC region, with both current and next generation of telescopes. In particular, the next Cherenkov Telescope Array will be able to scan the GC region with improved flux sensitivity and angular resolution.