From the Late Noachian period, through the Hesperian, and into the Amazonian periods on Mars, large outflow channels were formed. Many are interpreted to have originated through the catastrophic discharge of groundwater from martian aquifers, involving the release of up to millions of cubic-kilometers of water. Such a mechanism for outflow channel formation requires that martian aquifers were supplied with significant quantities of water some time prior to the discharge events. Typical groundwater recharge occurs due to the infiltration of surficial waters through a permeable substrate down into aquifers. However, some climate models predict an early martian climate dominated by generally “cold and icy” conditions. In this scenario, a globally continuous, impermeable cryosphere prevents infiltration of liquid water (that might be generated at the surface through anomalous heating conditions), leaving the martian aquifers without an apparent source of recharge to supply later outflow channel formation by groundwater discharge. More recent global climate modeling of an early, thicker CO2 martian atmosphere predicts that, when coupled with a full water cycle, the atmosphere of Mars will behave adiabatically causing temperatures to decrease with elevation. The high standing areas of Mars, such as the southern highlands and the Tharsis region, then act as cold traps. This leads to the preferential accumulation of snow and ice, resulting in the formation of regional ice sheets throughout the highlands that characterize the Late Noachian “icy highlands” early Mars climate model (LNIH). We make the initial assumption that the LNIH model is representative of the early Mars climate, and seek to test the model against the presence of the Hesperian and Amazonian outflow channels to determine if it can be consistent. In order to reconcile the LNIH early Mars climate model with the presence of the later outflow channels a groundwater recharge mechanism that can operate under the predicted “cold and icy” conditions is required. We test basal melting of surface snow and ice in response to a regionally elevated geothermal heat flux throughout the Tharsis rise (resulting from widespread volcanic and magmatic activity during the Noachian) as a mechanism that can provide: (1) liquid water generation at the surface of Mars under generally “cold and icy” conditions, and (2) potentially large scale integration of the hydrological system (through thinning or breaching of the cryosphere), allowing for infiltration of meltwater to provide groundwater recharge during the Late Noachian to supply the later formation of outflow channels. We find: (1) Regional scale basal melting of LNIH ice sheets is not likely to occur at the predicted nominal average ice sheet thicknesses, even in the presence of the anomalous bottom-up heating conditions expected in the Tharsis region (although the increased baseline heating will render the LNIH ice sheets more susceptible to melting through additional anomalous heating conditions introduced by top-down and bottom-up processes). (2) Local scale basal melting and groundwater recharge through a “heat-pipe drain pipe” mechanism is likely to occur, but is not predicted to produce sufficient groundwater recharge to supply the water needed to form the outflow channels. (3) Under the assumption of an ice saturated cryosphere, regional scale melting of the cryosphere due to the insulating effect of the LNIH ice sheets does not provide enough water to explain the formation of all of the outflow channels. Therefore, if the LNIH model is correct, the groundwater recharge that supplied outflow channel formation requires a source that operated earlier in martian history, or the recharge was supplied by other mechanisms.