The El Niño Southern Oscillation (ENSO) is a pattern of year-to-year variability in the Pacific ocean and overlying atmosphere. The Indian Ocean Dipole (IOD) is a similar phenomenon in the neighboring Indian Ocean. Both strongly impact global weather and climate. IOD and ENSO events frequently co-occur, and ENSO is known to influence the IOD. However, the role of the IOD in this relationship is still debated. Furthermore, the connection between ENSO and the IOD is stronger in some decades and weaker in others, and has recently weakened substantially; what causes this remains uncertain. In this study, we investigate changes in the strength of the IOD-ENSO connection in a global climate model run for the past 1000 years. We find that natural oscillations of the climate system can cause ENSO and the IOD to be more or less connected. Periods with less connection are associated with a stronger atmospheric circulation in the Pacific, which disrupts ENSO, changing its characteristics and its ability to connect with the IOD. In the future, under global warming, our results indicate that a weakening of the atmospheric circulation in the Pacific may lead to a stronger and more consistent connection between ENSO and the IOD.