Weakening of the Atlantic Meridional Overturning Circulation (AMOC) with greenhouse gas forcing is a common feature in most Earth System Models (ESMs). Here, we show that uncertainty in regional precipitation projections in CMIP6-class ESMs, particularly over the deep tropics, is linked to uncertainty in how much the AMOC weakens. Specifically, ESMs with greater AMOC weakening exhibit less strengthening (or greater weakening) of monsoonal circulations north of the equator, consistent with a pronounced southward shift in the tropical precipitation maximum. Using an idealized ESM, we show that decreasing the (northward) cross-equatorial ocean heat transport (X-OHT) while doubling CO2 is sufficient to shift the tropical precipitation maximum southward and thereby weaken northern hemispheric monsoon circulations, relative to an experiment where X-OHT remains unchanged. These findings imply that uncertainty in precipitation projections over key regions, particularly in the tropics, may be reduced by better constraining changes in large-scale ocean dynamics with warming.