Context. FUors outbursts are a crucial stage of accretion in young stars. However, a complete mechanism at the origin of the outburst still remains missing. Aims. We aim to constrain the instability mechanism in the star FU Orionis itself by directly probing the size and evolution in time of the outburst region with near-infrared (NIR) interferometry, and to confront it with physical models of this region. Methods. As the prototype of the FUors class of objects, FU Orionis has been a regular target of NIR interferometry. In this paper, we analyze more than 20 years of NIR interferometric observations to perform a temporal monitoring of the region of the outburst, and compare it to the spatial structure deduced from 1D magneto-hydrodynamic (MHD) simulations. Results. We measure from the interferometric observations that the size variation of the outburst region is compatible with a constant or slightly decreasing size over time: -0.56-0.36+0.14 AU/100 yr and -0.30-0.19+0.19 AU/100 yr in the H and K bands, respectively. The temporal variation and the mean size probed by NIR interferometry are consistently reproduced by our 1D MHD simulations. We find that the most compatible scenario is a model of an outburst occurring in a magnetically layered disk, where a magneto-rotational instability (MRI) is triggered by a gravitational instability (GI) at the outer edge of a dead zone. The scenario of a pure thermal instability (TI) fails to reproduce our interferometric sizes because it can only be sustained in a very compact zone of the disk <0.1 AU. The comparison between the data and the MRI-GI models favors MHD parameters of αMRI = 10−2, TMRI = 800 K, and Σcrit = 10 g cm−2, with more work needed in terms of observations and modeling in order to improve the precision of these values. Locally, in the very inner part of the disk, TI can be triggered in addition to MRI-GI, which qualitatively better matches our observation but is not strongly constrained by the currently available data. The scenario of MRI-GI could be compatible with an external perturbation that enhances the GI, such as tidal interaction with a stellar companion, or a planet at the outer edge of the dead zone. Conclusions. We favor a layered-disk model driven by MRI turbulence in order to explain the spatial structure and temporal evolution of the outburst region on FU Orionis. Understanding this phase will provide a crucial link between the early phase of disk evolution and the process of planet formation in the first inner astronomical units.