Monk joined the club in 2004 and was made skipper two years later but he has decided to allow Wales international Williams to take over the matchday duties.
The 34-year-old will remain as Swansea club captain but after making only 15 appearances last season, he has made way for Williams, who missed just one Premier League game in 2012/13.
Centre-back Williams will lead the team on the field, even when both men are playing.
"It's a personal decision from myself. The way I look at being a captain, you should consistently be on the pitch," Monk told the club's official website.
"Ash has been consistent on the pitch over a number of years. I've said a few times I've not played as much as I would have liked, but I've still contributed in a number of games.
"But, overall, I feel the team captain should be someone who is on there as a figure of continuity. Ash is that.
"He has worn the armband when I've not played, so I felt it was the right thing to do to let him lead the team out even when I was playing.
"Obviously, if the manager (Michael Laudrup) says anything different, then we would listen to him. But I will remain as club captain. To have been captain of
this team and club is one of my proudest achievements.
"My duties as club captain on the pitch, in the changing rooms and in the community will remain.
"Even when I haven't got the armband on when I'm playing, I'll still be a leader on the pitch."