As the greenhouse gas concentrations increase, a warmer climate is expected. However, numerous internal climate processes can modulate the primary radiative warming response of the climate system to rising greenhouse gas forcing. Here the particular internal climate process that we focus on is the Atlantic meridional overturning circulation (AMOC), an important global-scale feature of ocean circulation that serves to transport heat and other scalars, and we address the question of how the mean strength of AMOC can modulate the transient climate response. While the Community Earth System Model version 2 (CESM2) and the Energy Exascale Earth System Model version 1 (E3SM1) have very similar equilibrium/effective climate sensitivity, our analysis suggests that a weaker AMOC contributes in part to the higher transient climate response to a rising greenhouse gas forcing seen in E3SM1 by permitting a faster warming of the upper ocean and a concomitant slower warming of the subsurface ocean. Likewise the stronger AMOC in CESM2 by permitting a slower warming of the upper ocean leads in part to a smaller transient climate response. Thus, while the mean strength of AMOC does not affect the equilibrium/effective climate sensitivity, it is likely to play an important role in determining the transient climate response on the centennial time scale.