Climate change has an adverse effect on the global hydrological cycle and hence climatic extremes are becoming more persistent. Therefore, the present study analyses the characteristics of the compound extremes in India using the Godavari River basin as a test bed. The study examines the individual return period (IRP) and pairwise joint return period (JRP) of several climate extreme indices such as consecutive dry days (CDD), consecutive wet days (CWD), percentage rainfall of very dry days (R5Tot) and very hot days (TX95) considering historical and future period of two scenarios, SSP245 and SSP585, from the CMIP6 consortium. Both SSPs indicates an upward trend in the indices for CWD and TX95, whilst showcasing a declining trend for CDD and R5Tot at 5% significance level. Under SSP585, the values of JRPs , , and ranges from 3 to 5 years, 14–55 years, 35–274 years and 104 to >1000 during epoch‐1 (2023–2060). During epoch‐2 (2061–2100), the JRPs values are generally higher. Regions displaying lower JRPs values suggest a higher likelihood of simultaneous occurrences of extremes such as floods and droughts, as well as very dry days and warm days within the same year. Under SSP585, the maximum (minimum) value of change rate (CR) of JRP for and during epoch‐1 is more than 900% (−80%) and 1000% (−90%), whereas during epoch‐2, the corresponding CR values are greater than 700% (−80%) and 1000% (−90%). The future possibility of co‐occurrence of ‘CDD and CWD’ and ‘R5Tot and TX95’, both exceeding their 10‐ and 25‐year IRP, in a single year, is highly likely for the regions with low CR values and unlikely for the regions with high CR values. The study would help in understanding the spatial and temporal variability of compound climate extremes over the Godavari River basin.