Despite the unavoidable presence of silicon atoms in the catalyst alloy droplets during the vapor-liquid-solid growth of III-V nanowires on silicon substrates, it remains unknown how the nucleation of nanowires is affected by these foreign atoms. In this work, we present the first attempt to quantify the nanowire nucleation rate versus the silicon concentration in the droplet. We calculate the chemical potential difference per Ga-As pair in the quaternary Au-Ga-As-Si liquid alloy droplet and in solid state, and compare it to the ternary Au-Ga-As droplet without silicon. This allows us to compute the nucleation rates of GaAs nanowires versus the silicon concentration under different conditions. We find that the presence of silicon in the droplet decreases the nucleation probability of GaAs nanowires for gallium-rich droplets (with the gallium contents c(GA) greater than 0.6) and increases it for gold-rich droplets (c(GA) < 0.6). The model is used to explain our experimental data for hydride vapor phase epitaxy of gold-catalyzed GaAs nanowires, which easily nucleate on Si(111) covered with different SiO2 layers but do not grow on the bare Si(111). In the latter case, more silicon is etched from the substrate and enters the gallium-rich droplets, which suppresses the nanowire nucleation. We discuss other relevant data, including the known difficulties in obtaining self-assisted GaAs NWs on silicon by chemical epitaxy techniques. These results may be useful for the fine-tuning of III-V nanowire properties and integrating them with silicon electronics.