Rush said: "It's been 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.
"That's why I made my decision to stay and try to establish my managerial credentials at the Deva Stadium.
"I was given permission by my club to talk to the Welsh FA, and nothing would make me prouder than managing my country one day. At the same time, I feel very passionately about continuing my career at Chester. Leaving was the last thing on my mind. The recent developments really did come out of the blue.
"The timing of such an opportunity wasn't ideal, but football has a habit of throwing these kinds of things your way.
"I made sure I didn't rush my decision and having a great relationship with my chairman helped me make the commitment to stay.
"My main focus remains helping Chester move towards the play-off places and I intend to be the manager at the Saunders Honda Stadium for a long time to come."