Positive real conditions and differential sector conditions have recently been shown to imply global convergence w.p. 1, for recursive identification schemes based on a class of single-input/single-output nonlinear Wiener models. The models consist of linear dynamics followed by a static output nonlinearity. The model structure is hence closely related to that of the Lure problem in the stability theory of feedback systems. This paper proves that the conditions for convergence can be transformed to graphical circle criteria, depending on the sector conditions and on the Nyquist plot of a transfer function related to the prior knowledge of the poles of the identified system.