a process and apparatus for manufacturing tires for vehicle wheels , includes building , on a toroidal support , a carcass structure including at least one carcass ply associated , at axially opposite end edges thereof , with annular anchoring structures , the building step of the carcass structure includes forming at least one reinforcing structure , operatively associated with the annular anchoring structures through deposition on the toroidal support of at least one reinforcing element along a predetermined curvilinear deposition path . the deposition of the at least one reinforcing element includes the steps of : cutting to size a piece of at least one reinforcing continuous band - like element fed by a feeding device along a predetermined feeding direction , deforming the piece obtained according to the curvature of a deposition part of the deposition path intended to receive the piece , and depositing the deformed piece at the deposition part .