The rewinding machine comprises: a first winding roller (11) and a second winding roller (13) defining a nip (15) through which said cores are inserted and through which said web material (N) is fed; and at least one pair of motorized engaging members (57), to engage the ends of a winding core and transmit a rotational movement to the core during at least part of the winding cycle of each log. The engaging members and the first and second winding roller are produced and disposed so that the log being formed is in contact with said first and said second winding roller. Moreover, a third winding roller (17) is provided, defining a winding space with said first and said second winding roller, said third winding roller being movable to allow increase and completion of winding of each log in said winding space.