Molecular dynamics simulation has been used to explore the nature of solvation dynamics for an excess electron in methanol and in water. We perform the analysis within the linear response theory and show that nonlinear corrections are small in bath cases. The response function characterizing solvent relaxation after electron photoexcitation and that following the subsequent nonradiative transition are modeled and found to behave very similarly in methanol, in contrast to water. For methanol, each is comprised of an extremely short Gaussian inertial component of small amplitude and a bi-exponential diffusive decay. A relatively fast similar to 1 ps exponential accounts for approximately half of the solvent relaxation and is followed by a slower similar to 7 ps relaxation of comparable magnitude, a solvation response that is rather similar to that reported previously for relatively large molecules in methanol. Spectral densities of energy gap fluctuations for the equilibrium ground and excited state trajectories show that translational motion dominates solvation. Relaxational processes in methanol have been compared with the results for water. In contrast to methanol, librational motions of solvent molecules significantly influence aqueous solvation dynamics, especially following excited state decay. This difference is reflected in the relaxational processes, which are an order of magnitude slower in methanol than in water.