The machine comprises: a spindle (1); a winding unit (5), cooperating with the spindle to wind one or more strips of web material about the spindle and form a tubular element (T); a cutting unit (31) with at least one discoidal cutting tool to divide the tubular element into tubes of a length which can be predetermined. At least one pressure element is associated with the cutting tool, to compress the tubular element between the spindle (1) and the pressure element adjacent to the cut performed by the cutting tool.