Despite reported interest from Arsenal during the close-season, Huw Jenkins revealed that there have been no offers for Williams, or any of other Swansea players.
The 28-year-old England international joined the club permanently in 2008 and was part of their rise to the English top flight.
As a result, Jenkins is determined to keep hold of the player.
"We have had no phone calls about any of our players and we hope that continues," he told Sky Sports.
"Ash has grown with us. His performances and personality have grown with the club and he is a vital part of our team.
"He is a big part of our defence and now he is captain as well. He is a good leader to have at the club.
"As always, we want to keep those players and we hope he is with us for many years to come."
Swansea get their Premier League campaign under way on Saturday against Manchester United.
Michael Laudrup's side then welcome Petrolul in the UEFA Europa League play-offs on August 22.
