The 29-year old centre-back has been with the Welsh club since 2008 and Monk, who is waiting to find out if he will remain as boss next season, wants Williams to stay at the Liberty Stadium longer than the next 12 months of his contract allows him to.
Monk feels Williams’ service has helped build a culture of loyalty in south Wales and has implored him to see out his playing days with the Swans.
"It's not a situation I can try to resolve until I know whether I am doing this job (next season)," Monk said.
"But for me, Ash should finish his career here, and the club should do all they can to make him a priority.
"Whoever the manager is, Ash should finish his career here because he is an integral part of what this club stands for.
"The likes of him and Leon (Britton), people like that, are an integral part of Swansea City.
"They are what the club has been built on and they are the image of this club.
"For me they are some of the most important players this club have ever had, and we have to make sure they are here pushing their values on to the other players, and all the new players who come in, for years to come."
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