The East Greenland Current (EGC) had long been considered the main pathway for the Denmark Strait overflow (DSO). Recent observations, however, indicate that the north Icelandic jet (NIJ), which flows westward along the north coast of Iceland, is a major separate pathway for the DSO. In this study a two-layer numerical model and complementary integral constraints are used to examine various pathways that lead to the DSO and to explore plausible mechanisms for the NIJ’s existence. In these simulations, a westward and NIJ-like current emerges as a robust feature and a main pathway for the Denmark Strait overflow. Its existence can be explained through circulation integrals around advantageous contours. One such constraint spells out the consequences of overflow water as a source of low potential vorticity. A stronger constraint can be added when the outflow occurs through two outlets: it takes the form of a circulation integral around the Iceland–Faroe Ridge. In either case, the direction of overall circulation about the contour can be deduced from the required frictional torques. Some effects of wind stress forcing are also examined. The overall positive curl of the wind forces cyclonic gyres in both layers, enhancing the East Greenland Current. The wind stress forcing weakens but does not eliminate the NIJ. It also modifies the sign of the deep circulation in various subbasins and alters the path by which overflow water is brought to the Faroe Bank Channel, all in ways that bring the idealized model more in line with observations. The sequence of numerical experiments separates the effects of wind and buoyancy forcing and shows how each is important.