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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. Plain Language Summary: Melting of the Greenland Ice Sheet has been accelerating in recent decades because of rising ocean and air temperatures. Warm ocean water in the deep basin from the subtropical North Atlantic is separated from the ice sheet margin (glacier termini in the Greenland fjords) by the shallower continental shelf region. In this study we compare two simulations of the ocean and sea ice that represent the currents and eddying motions around Greenland realistically. We identify how and where heat is moved on and off the southern Greenland shelf and consider the results to be robust when they are common to both simulations. Warm water mainly moves onto the southeast shelf and off the southwest shelf; the currents on the shelf transport the warm water around the southern tip of Greenland. Near the Denmark Strait we identify oscillations in the warm water crossing onto the shelf that are associated with the presence of Denmark Strait Overflow eddies. On average, these eddies move heat onto the shelf in one model and off the shelf in the other. Understanding how warm water reaches the shelf allows us to better understand how the ocean contributes to the melting of the Greenland Ice Sheet. Key Points: Cross‐shelf heat flux is strongest over the southeast continental shelf in both POP and HYCOM ocean modelsDenmark Strait Overflow eddies traveling along the shelf break drive multi‐day oscillations of on‐shelf heat fluxOn‐shelf heat fluxes along the wide sector of the southeast Greenland shelf are associated with the mean flow in HYCOM and eddies in POP [ABSTRACT FROM AUTHOR]