We show that by illuminating an InGaAs/GaAs self-assembled quantum dot with circularly polarized light, the nuclei of atoms constituting the dot can be driven into a bistable regime, in which either a thresholdlike enhancement or reduction of the local nuclear field by up to 3 T can be generated by varying the pumping intensity. The excitation power threshold for such a nuclear spin "switch" is found to depend on both the external magnetic and electric fields. The switch is shown to arise from the strong feedback of the nuclear spin polarization on the dynamics of the spin transfer from electrons to the nuclei of the dot.