Williams, who replaced Garry Monk as team captain this summer, is thought to have been on Arsene Wenger's radar after impressing for the Welsh club since their promotion to the Premier League two years ago.
The Swans were said to have put a £10m price tag on Williams, 28, to put off any potential suitors, but Jenkins is determined that his player will stay at the Liberty Stadium.
"Ash has grown with us. His performances and personality have grown with the club and he is a vital part of our team," Jenkins declared.
"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."
Arsenal are yet to make any moves in the summer transfer market, and Jenkins is adamant that no approach has been made for Williams.
"We have had no phone calls about any of our players," said the Swansea chairman, "and we hope that continues."