A process for manufacturing a semiconductor pressure-monitoring device, comprising the steps of: - providing a wafer of monocrystalline silicon; - forming, in a first region of said wafer a first buried cavity and a first membrane, integrated in said wafer, suspended over, and closing at the top, said first buried cavity; - forming, in a second region of said wafer, a second buried cavity and a second membrane, integrated in said wafer, suspended over, and closing at the top, said second buried cavity; - depositing an inertial mass on top of a surface of said first membrane opposite to said first buried cavity, said inertial mass being rigidly coupled to said first membrane; - providing, in said first membrane, first piezoresistive transduction elements sensitive to strains of said first membrane due to movements of said inertial mass in response to a sensed acceleration and generating first electrical signals corresponding to the sensed acceleration, so as to provide an acceleration sensor; and - providing, in said second membrane, second piezoresistive transduction elements sensitive to strains of said second membrane in response to a sensed pressure and generating second electrical signals corresponding to the sensed pressure, so as to provide a pressure sensor integrated with said acceleration sensor in said wafer; wherein said steps of forming a first and a second buried cavity comprise: - digging, in said first and, respectively, second regions of said wafer, trenches delimiting between them walls of semiconductor material; MAIN REQUEST - epitaxially growing, starting from said walls, a closing layer of semiconductor material, said closing layer closing said trenches at the top; and - carrying out a thermal treatment such as to cause migration of the semiconductor material of said walls and form said first buried cavity and concurrently said first membrane in said first region, and said second buried cavity and concurrently said second membrane in said second region. - providing a wafer of monocrystalline silicon; - forming, in a first region of said wafer a first buried cavity and a first membrane, integrated in said wafer, suspended over, and closing at the top, said first buried cavity; - forming, in a second region of said wafer, a second buried cavity and a second membrane, integrated in said wafer, suspended over, and closing at the top, said second buried cavity; - depositing an inertial mass on top of a surface of said first membrane opposite to said first buried cavity, said inertial mass being rigidly coupled to said first membrane; - providing, in said first membrane, first piezoresistive transduction elements sensitive to strains of said first membrane due to movements of said inertial mass in response to a sensed acceleration and generating first electrical signals corresponding to the sensed acceleration, so as to provide an acceleration sensor; and - providing, in said second membrane, second piezoresistive transduction elements sensitive to strains of said second membrane in response to a sensed pressure and generating second electrical signals corresponding to the sensed pressure, so as to provide a pressure sensor integrated with said acceleration sensor in said wafer; wherein said steps of forming a first and a second buried cavity comprise: - digging, in said first and, respectively, second regions of said wafer, trenches delimiting between them walls of semiconductor material; MAIN REQUEST - epitaxially growing, starting from said walls, a closing layer of semiconductor material, said closing layer closing said trenches at the top; and - carrying out a thermal treatment such as to cause migration of the semiconductor material of said walls and form said first buried cavity and concurrently said first membrane in said first region, and said second buried cavity and concurrently said second membrane in said second region.