Analytical expressions are derived for the rates of longitudinal and transverse nuclear spin relaxation under conditions of fast modulation of the magnitude and direction of a hyperfine field induced by unpaired electrons of an ion. The results obtained are used to explain the data available in the literature on the 55Mn spin relaxation in the ferromagnetic metallic phase of doped perovskites, in which the modulation of the hyperfine field is caused by the hopping of eg electrons between Mn3+ and Mn4+ ions. It is demonstrated that, within this model, the rates of longitudinal and transverse relaxation are characterized by the same temperature dependence and their ratio is independent of temperature, which is in agreement with the experimental data.