The initial stage of hydrogen-metal reactions leading to the formation of metallic hydrides, frequently proceeds (on massive metallic samples) by the appearance of hydride "spots" (or "patches") on the surface of the metal. These spots (henceforth denoted as growth centers, GC's) grow at a certain velocity, overlap, and finally cover the whole surface by a hydride layer (which continues to thicken into the bulk of the metal). In certain systems it has experimentally been verified that the total available number of such GC's is limited to a certain fixed number (typical to the given metallic surface and the reaction conditions). The kinetics of formation of these GC's is characterized by a rate function which depends on the reaction conditions, i.e., temperature and pressure, and on the metal surface characteristics. Qualitatively, these rate functions initially accelerate, reach a maximum rate, and finally decay to zero. In the present article, a kinetic theory, which accounts for that behavior, is proposed. The model assumes that the initiation of the growth process can take place only at certain sites, with a limited number of such sites available on the surface. Also, the probability for the start-up of the growth process increases with increasing supersaturation of the hydrogen concentration at these sites. The interplay between these two factors (i.e., available GC's sites and local hydrogen concentrations) reproduces the observed kinetics of formation of the GC's. Analytical expressions for these rate functions are derived and compared with the experimental kinetic data reported for some metal-hydrogen systems. Universal forms of the rate function and its parameters provide a simple procedure for the evaluation of specific kinetic parameters from experimentally available data.