There are provided an EUV optical member, in which deterioration in the reflectivity due to oxidation of the Ru protective layer is prevented, a functional film-equipped substrate to be employed for production of the EUV optical member, and a process for producing the functional film-equipped substrate. A reflective layer-equipped substrate for EUV lithography comprising a substrate, and a reflective layer for reflecting EUV light and a protective layer for protecting the reflective layer, formed in this order on the substrate, wherein the reflective layer is a Mo/Si multilayer reflective film, the protective layer is a Ru layer or a Ru compound layer, and an intermediate layer containing from 0.5 to 20 at % of oxygen and from 80 to 99.5 at % of Si is formed between the reflective layer and the protective layer.