A machine for forming spools of web material, comprising: an unwinding section for unwinding parent reels (Ba, Bb) of web material (Na, Nb); a cutting station arranged downstream of the unwinding section and comprising cutting members adapted to divide the web material coming from the unwinding section into longitudinal strips; a plurality of winding stations arranged in sequence downstream of the cutting station, each of which is adapted to receive a respective longitudinal strip; wherein each winding station comprises a winding device (41, 51), to which the longitudinal strip of web material is fed, and in which a respective spool of web material is formed; the winding device (41, 51) being arranged and controlled to wind the longitudinal strip and to form the spool by rotation around a rotation axis (C-C) of the spool; wherein each winding device comprises a winding mandrel provided with a rotation movement around the rotation axis (C-C) and with a reciprocating translation movement in a direction parallel to the rotation axis (C-C), so as to helically wind the longitudinal strip around the winding mandrel; characterized in that each winding station further comprises: a strip cutting device (81, 83) for cutting the longitudinal strip transversally, thus forming a trailing edge, which remains on the spool under formation, and a leading edge; a holding device for holding the leading edge of the longitudinal strip, to transfer the leading edge onto a tubular winding core of a new spool to be wound; a contact roller associated with the winding mandrel, the contact roller rotating around a rotation axis and being adapted to be pressed against the outer surface of a spool being formed on the winding mandrel; and wherein the contact roller and the holding device are carried by a pivoting arm provided with a pivoting movement in order to move towards and away from the axis of the winding mandrel; and said pivoting arm carries at least a part of the cutting device (81, 83). an unwinding section for unwinding parent reels (Ba, Bb) of web material (Na, Nb); a cutting station arranged downstream of the unwinding section and comprising cutting members adapted to divide the web material coming from the unwinding section into longitudinal strips; a plurality of winding stations arranged in sequence downstream of the cutting station, each of which is adapted to receive a respective longitudinal strip; wherein each winding station comprises a winding device (41, 51), to which the longitudinal strip of web material is fed, and in which a respective spool of web material is formed; the winding device (41, 51) being arranged and controlled to wind the longitudinal strip and to form the spool by rotation around a rotation axis (C-C) of the spool; wherein each winding device comprises a winding mandrel provided with a rotation movement around the rotation axis (C-C) and with a reciprocating translation movement in a direction parallel to the rotation axis (C-C), so as to helically wind the longitudinal strip around the winding mandrel; characterized in that each winding station further comprises: a strip cutting device (81, 83) for cutting the longitudinal strip transversally, thus forming a trailing edge, which remains on the spool under formation, and a leading edge; a holding device for holding the leading edge of the longitudinal strip, to transfer the leading edge onto a tubular winding core of a new spool to be wound; a contact roller associated with the winding mandrel, the contact roller rotating around a rotation axis and being adapted to be pressed against the outer surface of a spool being formed on the winding mandrel; and wherein the contact roller and the holding device are carried by a pivoting arm provided with a pivoting movement in order to move towards and away from the axis of the winding mandrel; and said pivoting arm carries at least a part of the cutting device (81, 83).