The Swans' manager believes the 29-year-old wants to stay at the Liberty Stadium despite interest from Arsenal and Napoli while he also believes Wilfried Bony will not be leaving
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Williams has one year left on his contract at the Liberty Stadium and has been linked with moves to Arsenal, Napoli and Hull City ahead of next term.
However, manager Monk confirmed on Wednesday that Swansea have begun negotiations with the 29-year-old over a new contract to keep him at the club.
"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," Monk told the South Wales Evening Post.
"I know Ash wants that, I want that, the club wants that and that's what we're working towards as we speak."
Monk, who was named permanent manager at the end of last season, also expects forward Wilfried Bony to remain at the club, even though other clubs are reported to be interested in a move for the Ivory Coast international.
"I spoke to him (Bony) 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," Monk added.
"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."