The steady incompressible Navier–Stokes equations in a two-dimensional exterior domain are considered with a nonzero constant velocity at infinity. It is known that the asymptotic behavior of the velocity field is described by the linearization around the velocity at infinity. I will present the asymptotic behavior of the vorticity and explain why surprisingly it is not characterized by the previous linearization. More precisely, the vorticity has a power of decay at infinity which depends continuously on the data.