Aims. It has been suggested that X-ray observations of rapidly variable Seyfert galaxies may hold the key to probe the gas orbital motions in the innermost regions of accretion discs around black holes and, thus, trace flow patterns under the effect of the hole strong gravitational field. Methods. We explore this possibility by re-analyzing the multiple XMM-Newton observations of the Seyfert 1 galaxy NGC 3783. A detailed time-resolved spectral analysis is performed down to the shortest possible time-scales (few ks) using excess maps and cross-correlating light curves in different energy bands. Results. In addition to a constant core of the Fe Kα line, we detected a variable and redshifted Fe Kα emission feature between 5.3-6.1 keV. The line exhibits a modulation on a time-scale of ∼27 ks that is similar to, and in phase with, a modulation of the 0.3-10 keV source continuum. The two components show a good correlation. Conclusions. The time-scale of the correlated variability of the redshifted Fe line and continuum agrees with the local dynamical time-scale of the accretion disc at ∼10 rg around a black hole with the optical reverberation mass ∼10 7 M ⊙ . Given the shape of the redshifted line emission and the overall X-ray variability pattern, the line is likely to arise from the relativistic region near the black hole, although the source of the few cycles of coherent variation remains unclear.