Marine heatwaves are becoming more frequent, widespread and severe1, leading to mass coral bleaching and mortality. Yet it remains unknown whether natural coral adaptation can keep pace with climate warming2. As a result, selective breeding has been proposed to enhance coral heat tolerance3. The viability of this management solution hinges on the extent of heritability of coral heat tolerance. We show that selecting parent colonies for heat tolerance results in higher tolerance of adult offspring. This result held true for the response to both a 1-week and a 1-month simulated marine heatwave. In each case, narrow-sense heritability (h2) estimates are between 0.2 and 0.3, demonstrating a substantial genetic component to variation in heat tolerance. Given the variability in this population4, selective breeding could enhance heat tolerance by up to 1 °C-week within one generation. Contrary to expectations, we found no genetic correlation between short- and long-term heat stress tolerance. Selective breeding for short-stress tolerance did not enhance the ability of offspring to survive the long heat stress exposure, demonstrating that these traits are subject to independent genetic control. We demonstrate that selective breeding to enhance coral heat tolerance is feasible and a potential solution to help corals face ocean warming.