Two proposals are made for producing nuclear polarization by a transient analog of the Overhauser effect in the impurity-state electron spin resonance in silicon. The success of the proposal polarization schemes requires that the electron spin relaxation process in which the impurity nucleus and the electron simultaneously flip their spins be faster than those relaxation processes in which an electron alone changes its spin. A detailed study of many electron spin relaxation processes is carried out. It is found that for sufficiently low concentrations of P, As, and Sb impurities in silicon transient nuclear polarization should occur; for Li impurities it should not. The foregoing theoretical predictions have been verified for As by the recent experiments of Abragam and Combrisson, and of Feher.