The basal thermal state of an ice sheet (frozen or thawed) is an important control upon its evolution, dynamics and response to external forcings. However, this state can only be observed directly within sparse boreholes or inferred conclusively from the presence of subglacial lakes. Here we synthesize spatially extensive inferences of the basal thermal state of the Greenland Ice Sheet to better constrain this state. Existing inferences include outputs from the eight thermomechanical ice-flow models included in the SeaRISE effort. New remote-sensing inferences of the basal thermal state are derived from Holocene radiostratigraphy, modern surface velocity and MODIS imagery. Both thermomechanical modeling and remote inferences generally agree that the Northeast Greenland Ice Stream and large portions of the southwestern ice-drainage systems are thawed at the bed, whereas the bed beneath the central ice divides, particularly their west-facing slopes, is frozen. Elsewhere, there is poor agreement regarding the basal thermal state. Both models and remote inferences rarely represent the borehole-observed basal thermal state accurately near NorthGRIP and DYE-3. This synthesis identifies a large portion of the Greenland Ice Sheet (about one third by area) where additional observations would most improve knowledge of its overall basal thermal state.