A tool-holding spindle (1) for machining with the removal of swarf in which rotary movement of the spindle (1) and a tool (2) coaxially fixed to it and axial forward and backward movement of the tool (2) are required, in which the tool (2) is fixed in a hollow member (7) which can move axially and rotate with respect to the body (1d) of the spindle (1) within which it is housed, magnets (4n) forming part of a linear motor (4) whose stator windings (4s) are housed in the body (1d) of the said spindle (1) around the hollow member (7) being fixed to the said hollow member (7), is described.