The spatial distribution of basal water critically impacts the evolution of ice sheets. Current estimates of basal water distribution beneath the Greenland Ice Sheet (GrIS) contain large uncertainties due to poorly constrained boundary conditions, primarily from geothermal heat flux (GHF). The existing GHF models often contradict each other and implementing them in numerical ice-sheet models cannot reproduce the measured temperatures at ice core locations. Here we utilize two datasets of radar-detected basal water in Greenland to constrain the GHF at regions with a thawed bed. Using the three-dimensional ice-sheet model SICOPOLIS, we iteratively adjust the GHF to find the minimum GHF required to reach the bed to the pressure melting point, GHF_pmp, at locations of radar-detected basal water. We identify parts of the central-east, south and northwest Greenland with significantly high GHF_pmp. Conversely, we find that the majority of low-elevation regions of west Greenland and parts of northeast have very low GHF_pmp. We compare the estimated constraints with the available GHF models for Greenland and show that GHF models often do not honor the estimated constraints. Our results highlight the need for community effort to reconcile the discrepancies between radar data, GHF models, and ice core information.