Wire diameter tolerance is set by standards such as EN 10218-2 (general wire), EN 10270-3 (spring wire), ASTM A555 (general) and ASTM A313 (spring). Tolerance classes tighten as diameter reduces; fine wire is held to a few microns.
Ovality, the difference between largest and smallest diameter of the same cross-section, is usually limited to half the total tolerance. For weaving, springing and cold heading it matters as much as diameter.
Specify diameter, tolerance class, ovality and the standard on the order, plus coil or spool weight and inside diameter, because these set what the mill quotes and what your machine accepts.