The 29-year-old, who has been at the Welsh side since 2008, has been one of the Swans' most influential players in their rise to the Premier League.
Williams will have one year remaining on his current deal at the end of the season, with Arsenal among the teams said to be interested in his services.
"It's not a situation I can try to resolve until I know whether I am doing this job [next season]," Monk is quoted as saying by the South Wales Evening Post on Williams's contract situation.
"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."
Williams has featured 39 times for Swansea this season.