Wales international Williams has been a reported target for Arsenal and QPR.
But Swansea boss Garry Monk remains hopeful that both Williams and World Cup-bound striker Wilfried Bony will stay.
"We are in negotiations with his (Williams') representatives and trying to put a deal across so that Ash will be here for the future of the club," Monk told the South Wales Evening Post.
"I know Ash wants that, I want that, the club wants that, and that's what we are working towards as we speak."
As for Ivory Coast forward Bony, Monk added: "I spoke to him at the end of the season and he knows how I feel, he knows how the club feels, he knows what I expect from him.
"I just want him to concentrate on the World Cup, have a good World Cup and he will be coming back as a Swansea City player."