The Arsenal manager is reported to have been handed a £100m war chest ahead of signing a three-year contract extension.
And Milner, 28, is one of the players Wenger has high up on his summer wishlist as he hopes to add competition for the likes of Mikel Arteta and Mathieu Flamini.
Milner has made it clear that he will be available this summer after a season of limited opportunities at the Etihad, while City’s £49m fine for breaking UEFA’s Financial Fair Play regulations, which is thought to have affected Manuel Pellegrini’s transfer plans.
However, Wenger will have to fend off competition from the likes of Tottenham, Liverpool and Everton for Milner’s services, with all three clubs also reported to have made enquiries for the England international.
But a deal for any club may have to wait until after this summer’s World Cup, with Three Lions boss Roy Hodgson insisting that he will not allow any transfer activity to disrupt his preparations for Brazil.