There has been plenty of speculation about the future of the Swansea skipper, including reports linking him with Arsenal and QPR.
But manager Monk said today he was hopeful both the Wales centre-back, 29, and striker Wilfried Bony, 25, would remain at the Liberty Stadium.
He said: "We're in negotiations with his representatives and trying to put a deal across so that Ash will be here for the future of the club,"
"I know Ash wants that, I want that, the club wants that and that's what we're working towards as we speak."
Bony, who is in the Ivory Coast World Cup squad, has also been linked with a host of clubs including Arsenal, Liverpool and Everton.
But Monk expects him to be playing for Swansea next season.
"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'll be coming back as a Swansea City player.
"That's what I have in my mind unless something happens that we can't predict, but as of now he's a Swansea City player and that's the way it should be.
"Every player that is under contract at Swansea is a Swansea City player... no one will dictate to the club, at the end of the day we decide where our players go and what players stay.
"I know players will push for moves and do this and that, but ultimately the decision is with the club which is a good position to be in."