We use neural networks and large climate model ensembles to explore predictability of internal variability in sea surface temperature anomalies on interannual (1-3 year) and decadal (1-5 and 3-7 year) timescales. We find that neural networks can skillfully predict SST anomalies at these lead times, especially in the North Atlantic, North Pacific, Tropical Pacific, Tropical Atlantic and Southern Ocean. The spatial patterns of SST predictability vary across the nine climate models studied. The neural networks identify “windows of opportunity” where future SST anomalies can be predicted with more certainty. Neural networks trained on climate models also make skillful SST predictions in historical observations, although the skill varies depending on which climate model the network was trained. Our results highlight that neural networks can identify predictable internal variability within existing climate datasets and show important differences in how well patterns of SST predictability in climate models translate to the real world.