The onset of collisionless magnetic reconnection is considered to be controlled by electron dynamics in the electron diffusion region, where the reconnection electric field is balanced mainly by the off-diagonal electron pressure tensor term. Two-dimensional particle-in-cell simulations are employed in this paper to investigate the self-reinforcing process of the reconnection electric field in the electron diffusion region, which is found to grow exponentially. A theoretical model is proposed to demonstrate such a process in the electron diffusion region. In addition the reconnection electric field in the pileup region, which is balanced mainly by the electromotive force term, is also found to grow exponentially and its growth rate is twice that in the electron diffusion region.