The presence of warm Atlantic water on the Greenland continental shelf has been connected to the accelerated melting of the Greenland Ice Sheet, particularly in the southwest and southeast shelf regions. Results from two high-resolution coupled ocean-sea ice simulations that utilized either the 1/10° Parallel Ocean Program (POP) or the 1/12° HYbrid Coordinate Ocean Model (HYCOM) are used to understand the flux of heat on and off the southern Greenland shelf. The analysis reveals that the region of greatest heat flux onto the shelf is southeast Greenland. On the southwestern shelf, heat is mainly exported from the shelf to the interior basins. We identify differences in the shelf break current structure and on-shelf heat content between the two simulations. Just south of the Denmark Strait, there is a seasonally persistent pattern of multi-day variability in the cross-shelf heat flux in both simulations. In the POP simulation, this high-frequency signal results in net on-shore heat flux. In the HYCOM simulation, the signal is weaker and results in net off-shelf heat flux. This variability is consistent with Denmark Strait Overflow eddies traveling along the shelf break.