It is well known that the stiff van der Pol equation has a strongly attractive limit cycle. In this paper it is shown that the Euler method applied to the van der Pol equation with small step size, small compared to the perturbation parameter, admits an attractive invariant closed curve close to the limit cycle. To describe closed curves in the vicinity of the limit cycle, 14 charts are introduced. A general graph transform result is derived and applied in these charts. The proof of the main result relies on the contraction principle in a suitable function space. Estimates are given for the distance of the invariant curve to the limit cycle.