Abstract The recent imaging of the M87 black hole at millimeter wavelengths by the Event Horizon Telescope (EHT) collaboration has triggered a renewed interest in numerical models for the accretion of magnetized plasma in the regime of general relativistic magnetohydrodynamics (GRMHD). Here non-ideal simulations, including both the resistive effects and, above all, the mean-field dynamo action due to sub-scale, unresolved turbulence, are applied for the first time to such systems in the fully nonlinear regime. Combined with the differential rotation of the disk, the dynamo process is able to produce an exponential growth of any initial seed magnetic field up to the values required to explain the observations, when the instability tends to saturate even in the absence of artificial quenching effects. Before reaching the final saturation stage we observe a secondary regime of exponential growing, where the magnetic field increases more slowly due to accretion, which is modifying the underlying equilibrium. By varying the dynamo coefficient we obtain different growth rates, though the field seems to saturate at approximately the same level, at least for the limited range of parameters explored here, providing substantial values for the MAD parameter for magnetized accretion. For reasonable values of the central mass density and the commonly employed recipes for synchrotron emission by relativistically hot electrons, our model is able to reproduce naturally the observed flux of Sgr A*, the next target for EHT.