We consider the possibility of anisotropic heat flow in the inner core by examining the potential for anisotropic thermal conductivity of hexagonal close-packed (hcp-)Fe. Because hcp-Fe exists only at pressures above 13 GPa at room temperature, we investigate thermal conductivity anisotropy in analog material Gd by measuring the electrical conductivity and applying the Wiedemann–Franz Law to determine thermal conductivity (k). The electrical conductivity anisotropy of Gd was measured at pressures up to 1.4 GPa and temperatures up to 873 K in the hcp phase range. At elevated pressure, the variation with temperature of anisotropic thermal conductivity of Gd single crystal resembles the anisotropic behavior at high temperature and 1 atm observed in earlier work. The temperature range of anisotropy of thermal conductivity of Gd, where kc > ka, is extended by pressure, but the anisotropy disappears before the high temperature hcp→bcc (body-centered cubic) transformation. Our results on hcp-Gd lead us to raise the question of the possibility of hcp-Fe exhibiting anisotropy of thermal conductivity. Together with the known seismic anisotropy of the inner core, and the inferred textural alignment of hcp crystals causing it, we suggest some implications that an anisotropy of thermal conductivity of hcp-Fe, and a concomitant anisotropy of inner core heat flow, could have on thermally driven core processes.