A machine (100, 101) for winding a conductor wire around a toroidal core, to obtain a toroidal coil, comprising: a support adapted to stably accommodate a toroidal core, said support being rotatable around a central vertical axis Z; a hook shaped to grab the conductor wire and arranged to translate vertically, having a trajectory passing through the central opening of the toroidal core positioned on the support; a forming arm which is controllable in position on three axes, namely according to a set of three orthogonal axes comprising said vertical axis Z and a further pair of horizontal axes X, Y, said forming arm thus being freely positionable in space around the toroidal core fixed to the support; wherein said forming arm carries a wire guide member which is adapted to engage the conductor wire so as to drag the conductor wire therewith and fold it in space; wherein said machine has a control system configured to control in a coordinated manner the vertical position of the hook, the position in space of the forming arm, provided with said wire guide member, and the rotation angle of the rotating support to automatically wind coils of the conductor wire around the core, comprising at least one gripper (28, 29) located close to a housing of the core defined by jaws for clamping the core and adapted to clamp one end of a piece of conductor wire, to define an anchoring and a starting point of winding of the conductor wire, characterized in that said at least one gripper includes: a pair of grippers (28, 29) which are arranged above and on opposite sides with respect to the housing for the core, or a single gripper with mobility on two axes or preferably on three axes so that the gripper can be positioned on both sides of the core housing and forming arm. a support adapted to stably accommodate a toroidal core, said support being rotatable around a central vertical axis Z; a hook shaped to grab the conductor wire and arranged to translate vertically, having a trajectory passing through the central opening of the toroidal core positioned on the support; a forming arm which is controllable in position on three axes, namely according to a set of three orthogonal axes comprising said vertical axis Z and a further pair of horizontal axes X, Y, said forming arm thus being freely positionable in space around the toroidal core fixed to the support; wherein said forming arm carries a wire guide member which is adapted to engage the conductor wire so as to drag the conductor wire therewith and fold it in space; wherein said machine has a control system configured to control in a coordinated manner the vertical position of the hook, the position in space of the forming arm, provided with said wire guide member, and the rotation angle of the rotating support to automatically wind coils of the conductor wire around the core, comprising at least one gripper (28, 29) located close to a housing of the core defined by jaws for clamping the core and adapted to clamp one end of a piece of conductor wire, to define an anchoring and a starting point of winding of the conductor wire, characterized in that said at least one gripper includes: a pair of grippers (28, 29) which are arranged above and on opposite sides with respect to the housing for the core, or a single gripper with mobility on two axes or preferably on three axes so that the gripper can be positioned on both sides of the core housing and forming arm. a pair of grippers (28, 29) which are arranged above and on opposite sides with respect to the housing for the core, or a single gripper with mobility on two axes or preferably on three axes so that the gripper can be positioned on both sides of the core housing and forming arm.