Rush, whose managerial career extends to 11 games in charge of Chester, was given permisison to talk to the FAW yesterday.
But the 43-year-old rookie boss has instead committed his future to the Chesire club.
He said: "It's very flattering for me to be linked with the Welsh job since Mark Hughes left.
"But having just started at Chester I found myself in a very difficult position."
Rush was one of six men on a short-list of six for the job, with John Toshack the outstanding favourite.
Former Liverpool chief Gerard Houllier is also on the list, along with Dean Saunders, Dick Advocaat and Phillipe Troussier.