Reduced‐order dynamical models play a central role in developing our understanding of predictability of climate irrespective of whether we are dealing with the actual climate system or surrogate climate models. In this context, the linear inverse modeling (LIM) approach, by capturing a few essential interactions between dynamical components of the full system, has proven valuable in providing insights into predictability of the full system. We demonstrate that reservoir computing (RC), a form of learning suitable for systems with chaotic dynamics, provides an alternative nonlinear approach that improves on the predictive skill of the LIM approach. We do this in the example setting of predicting sea surface temperature in the North Atlantic in the preindustrial control simulation of a popular earth system model, the Community Earth System Model so that we can compare the performance of the new RC‐based approach with the traditional LIM approach both when learning data are plentiful and when such data are more limited. The improved predictive skill of the RC approach over a wide range of conditions—larger number of retained EOF coefficients, extending well into the limited data regime, etc.—suggests that this machine‐learning technique may have a use in climate predictability studies. While the possibility of developing a climate emulator—the ability to continue the evolution of the system on the attractor long after failing to be able to track the reference trajectory—is demonstrated in the Lorenz‐63 system, it is suggested that further development of the RC approach may permit such uses of the new approach in more realistic predictability studies. Plain Language Summary: Because of the chaotic nature of the dynamics underlying many complex systems such as weather and climate, evolution of ensembles of trajectories have to be considered in order to produce future predictions of such systems. An analysis of the dynamics of such ensembles provides insights into mechanisms that make the system predictable. Because comprehensive models of such complex systems are very costly to run, reduced‐order dynamical models of them are a useful tool in conducting such ensemble‐based predictability studies. We develop such a computationally inexpensive reduced‐order model using machine learning and show that its predictive skill is comparable to those of linear inverse model (state‐of‐the‐art) when training data are plentiful, but much better when such data are more limited. Consequently, we think that this new method has wide applicability. Furthermore, given the nonlinear nature of the new method, it has the potential to provide new insights into predictability of complex systems. Key Points: A machine‐learning (ML)‐based reduced‐order modeling method is developed to study predictability of regional SST of a popular climate modelThe method extends the state‐of‐the‐art, particularly, when data are limited and when a large number of EOF coefficients are retainedThe new method has wide applicability, could act as a climate emulator with further developments, and can give new insights [ABSTRACT FROM AUTHOR]