The Madden and Julian Oscillation (MJO) can affect ENSO via two dynamical regimes: by linear interactions where the low-frequency tail of MJO is amplified via non-normal processes, and by non-linear interactions where some of the intraseasonal variability is rectified to low-frequencies. In this work we study these two types of interactions by forcing a hierarchy of ENSO coupled models with two estimates of the MJO. The first estimate concentrates in the variability at intraseasonal frequencies. In the second estimate, we first remove the ENSO related anomalies with the aid of an empirical model, and then we seek for the same MJO structures of the first estimate at lower frequency bands. We show that the spatial structure that dominates the intraseasonal band (i.e., the MJO) also dominates the low frequency once the anomalies directly forced by SST have been removed (i.e., those identified by the empirical model). This low-frequency contribution, having the same shape as that of the MJO, is shown to have a large impact on the ENSO models. The impact of the first estimate, composed of variability just in the intraseasonal band, will be also discussed.