In the current study, the effect of small surface defects in the starting material including roughness, indentations, or scratches, which are perpendicular to the direction of drawing, on the product quality is investigated using the finite element method. An axisymmetric defect is assumed. Such defects are defined by a cylindrical defect area and two tapered regions connecting the defect area to the non-defective area of the material. Various conditions for these initial surface defects are considered, including defect depth, defect slope and defect length. To describe the plastic deformation of the defect in detail during the simulation, local remeshing is applied. Based on the finite element results, defect disappearance maps were generated. It was found that defect disappearance is significantly dependent on the defect depth and the defect length coupled with the defect slope.