Isotopically light (−1‰ to −8‰ ) and fresh pore fluids (300–520 mM Cl−) were found in continental shelf sediments up to 100 km offshore of southeastern (SE) Greenland, suggesting infiltration and mixing of ice sheet meltwater with seawater to depths of 260 m. These geochemical anomalies may be associated with ice sheet–derived submarine groundwater discharge (SMGD). We present a continental‐scale reconstruction of the late Pleistocene hydrogeology of SE Greenland using finite element analysis that incorporates ice sheet loading and solute and isotope transport. Results indicate that subglacial infiltration and SMGD are of the same order of magnitude and are highly dependent on the permeability of the subaerial basalt. Simulated infiltration and SMGD almost doubled during the Last Glacial Maximum, compared to ice‐free conditions. Much of the present‐day glacially induced groundwater discharge occurs on land. Subice infiltration on the continental shelf likely represents a mixture of seawater and meltwater during past glacial maximums. Simulated SMGD during the most recent interval of glacial retreat is about 4% of the total ice sheet melting. At present, the simulated rate of SMGD is about 11% of the estimated current melting rate.