Large multi-decadal fluctuations of El Niño-Southern Oscillation (ENSO) variability simulated in a 4000-year pre-industrial control run of GFDL CM2.1 have received considerable attention due to implications for constraining the causes of past and future changes in ENSO. We evaluated the mechanisms of this low-frequency ENSO modulation through analysis of the extreme epochs of CM2.1 as well as through the use of a linearized intermediate-complexity model of the tropical Pacific, which produces reasonable emulations of observed ENSO variability. We demonstrate that the low-frequency ENSO modulation can be represented by the simplest model of a linear, stationary process, even in the highly nonlinear CM2.1. These results indicate that CM2.1’s ENSO modulation is driven by transient processes that operate at interannual or shorter time scales. Nonlinearities and/or multiplicative noise in CM2.1 likely exaggerate the ENSO modulation by contributing to the overly active ENSO variability. In contrast, simulations with the linear model suggest that intrinsically-generated tropical Pacific decadal mean state changes do not contribute to the extreme-ENSO epochs in CM2.1. Rather, these decadal mean state changes actually serve to damp the intrinsically-generated ENSO modulation, primarily by stabilizing the ENSO mode during strong-ENSO epochs. Like most coupled General Circulation Models, CM2.1 suffers from large biases in its ENSO simulation, including ENSO variance that is nearly twice that seen in the last 50 years of observations. We find that CM2.1’s overly strong ENSO variance directly contributes to its strong multi-decadal modulation through broadening the distribution of epochal variance, which increases like the square of the long-term variance. These results suggest that the true spectrum of unforced ENSO modulation is likely substantially narrower than that in CM2.1. However, relative changes in ENSO modulation are similar between CM2.1, the linear model tuned to CM2.1, and the linear model tuned to observations, underscoring previous findings that relative changes in ENSO variance can robustly be compared across models and observations.