The line comprises a feed path (P) of the web material (N) and, along the feed path, a transverse cutter (5) for cutting scraps (FS) from the continuous web material (N). The line includes a system for removing scraps of web material cut by the cutter. The system for removing scraps is fitted with a scrap accumulator member (21). The accumulator member (21) is movable in a direction generally parallel to the feed path (P) of the web material between: a position for receiving scraps, wherein the distance between the accumulator member (21) and the cutter (5) is such that scraps (FS) generated by the cutter (5) are collected by the accumulator member (21); and a standby position, at a distance from the cutter (5) such that scraps (FS) generated by the cutter are collected in a space (D) between the cutter (5) and the accumulator member (21).