Due to its ease of application, the circle criterion has been widely used to guarantee the stability of many anti-windup schemes. While the Popov criterion gives less conservative results, it has been conjectured in the literature that it cannot be used for convex anti-windup synthesis. This paper shows that the conjecture does not necessarily apply in the discrete-time setting. We show how the search for optimal parameters corresponding to the Jury-Lee criterion (a discrete counterpart of the Popov criterion) can be formulated as a convex search via a linear matrix inequality (LMI). The result is then extended to two existing multivariable static anti-windup schemes with stable open-loop plants. Two numerical examples of multivariable anti-windup controller synthesis are provided, and it is shown that in both cases the synthesis using the Jury-Lee criterion can allow better performance than existing methods which use the circle criterion alone.