In the framework of the ab initio random structure search method, we show that elemental Se and Te undergo pressure-induced structural transition from the bcc to fcc phase, in agreement with the theoretical results previously reported. By means of the pseudopotential plane-wave method based on density functional perturbation theory, the fcc structure for both elements is found to be another phonon-mediated superconducting phase of these materials. With a reasonable value for the Coulomb pseudopotential 𝜇 * = 0.12, the maximum superconducting transition temperature 𝑇c in the fcc phase of Se and Te is estimated to be about 5.7 K and 4.6 K, respectively. Furthermore, we show that in the entire fcc phase for Se and Te, the superconducting transition temperature decreases together with the increase in pressure, leading to the final suppression of the superconductivity. It is suggested that such behavior is mainly caused by the rapid increase in the mean-square phonon frequency ⟨𝜔 2 ⟩ with pressure. Finally, a very strong electron-phonon coupling value, for both Se and Te in the fcc phase, is found along the 𝐺-𝐾 high symmetry lines.