Monk, who joined the Swans in 2004 and became skipper in 2006, made just 15 appearances last season.
The 34-year-old is to retain the club captaincy, but 28-year-old Williams - out of the side for only one Barclays Premier League game last term - will lead the team on the field now, even when the pair are playing together.
Monk told www.swanseacity.net: "It's a personal decision from myself. The way I look at being a captain, you should consistently be on the pitch.
"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."