A process for manufacturing a MEMS pressure sensor having a micromechanical structure envisages: providing a wafer having a substrate of semiconductor material and a top surface; forming a buried cavity entirely contained within the substrate and separated from the top surface by a membrane suspended above the buried cavity; forming a fluidic-communication access for fluidic communication of the membrane with an external environment, set at a pressure the value of which has to be determined; forming, suspended above the membrane, a plate region made of conductive material, separated from the membrane by an empty space; and forming electrical-contact elements for electrical connection of the membrane and of the plate region, which are designed to form the plates of a sensing capacitor, the value of capacitance of which is indicative of the value of pressure to be detected. A corresponding MEMS pressure sensor having the micromechanical structure is moreover described.