The main paradigm for prediction of Indian Summer Monsoon Rainfall (ISMR) is its inverse relation with El Niño-Southern Oscillation (ENSO). In this study, we focus on the role of the Atlantic Ocean (AO) Sea Surface Temperature (SST) variability on the ISMR. There are basically two ways by which AO SSTs can impact the ISMR: a "direct pathway" in which the AO may directly force the ISMR in the absence of interactions with other dominant forcings like ENSO, and an "indirect pathway" in which AO forces ENSO and modulates the ENSO teleconnection to ISMR. These two pathways are studied with the help of sensitivity experiments performed with a Coupled General Circulation Model. Two pairs of decoupling experiments have been done. In the first, the SST variability in the tropical AO or Pacific Ocean (PO) is removed by nudging the SST in these regions from a control run's SST climatology. In the second set, the SST nudging is performed from the observed SST climatology, which allows us to assess the robustness of the results and the specific role of the model's SST mean-state biases. The direct pathway linking tropical AO SST variability onto ISMR is insignificant in the PO decoupled experiments or in recent observations. Furthermore, these experiments suggest on the contrary that many AO SST anomalous patterns could be forced by ISMR. On the other hand, for the indirect pathway, the AO decoupled experiments demonstrate that AO SST variability modulates the onset and decaying phases of ENSO events. Despite ENSO is as strong and persists longer than in the control simulation, the AO SST nudging resulted in a significant weakening of the inverse relationship between ENSO and ISMR. The ENSO-monsoon relationship is mainly modulated during the ENSO decaying phase. The upper-level divergent wind flows mainly from the Pacific to the AO resulting in rainfall suppression in the AO, but only in a weak forcing on ISMR during boreal summer of the ENSO decaying year in the AO decoupled experiments. Thus, the AO rainfall variability in these experiments is decoupled from the surface and mainly modulated by the upper-level convergence or divergence induced by the remote ENSO forcing. Finally, the rectification of the AO SST mean-state biases in the CGCM also induces an El Niño-like mean pattern over the tropical Pacific during boreal spring and promotes a stronger ENSO during its peak phase. This demonstrates that the prominent AO SST mean-state biases in current CGCMs further complicate the dynamical prediction and simulation of ISMR and ENSO.