Williams has signed a four-year contract extension, despite being linked with a summer move to Arsenal and Sunderland, but Monk has insisted that the Wales international had always signalled his intent to remain at the club.
The 35-year-old told the South Wales Evening Post: "I've got a good relationship with Ash. He's got great affection for the club and he's wanted to stay here from day one.
"I'm just happy for him and happy the club has secured him again. He can now concentrate on being captain and the football side of things. He can go into pre-season with a clear head.
"Ash has given good service to the club, but he's a quality player. He's a big player for the club - let's not forget that."
Williams, who spent the early parts of his career with Hednesford Town and Stockport County, has made 273 appearances for the Swans since joining the club in 2008.