The rewinding machine includes a first winding roller, around which the web material is guided, and defining at least in part a winding cradle; a support surface of the winding cores arranged to receive a winding core and to convey the core toward the winding cradle, the support surface defining with the first winding roller a feed channel of winding cores, a severing member of the web material, which can be inserted in the channel to sever the web material, the severing member interacting with the web material to cause severing thereof; a motor for controlling the severing member, the motor controlling the severing member modifying the speed of the severing member when it is positioned inside the channel.