A process for manufacturing a reinforcing structure of a tire for vehicle wheels, is carried out on a toroidal support including an outer surface, by means of at least one laying element, which can be oriented in space in a controlled way; the laying element including a laying surface adapted to act on a portion of a rubberized strip-like element for laying the same on the outer surface of the toroidal support. The reinforcing structure includes at least a first and a second radially superimposed reinforcing layers, each including a plurality of strip-like elements; each reinforcing layer having a circumferential extension about a rotation axis of the tire and comprising thread-like reinforcing elements, arranged parallel to one another. The layers are arranged so that the reinforcing thread-like elements of the first reinforcing layer are oriented obliquely with respect to the equatorial plane of the tire, and that the reinforcing thread-like elements of the second reinforcing layer also have an oblique orientation, crossed with respect to the thread-like elements of the first layer.