The club chief revealed there have been no official bids for the defender despite rumours suggesting Arsenal want the Wales international at the Emirates
The 28-year-old, who was named team captain earlier this summer, has been persistently linked with Arsenal amidst rumours that the Swans had placed a £10m price-tag on his head.
However the Welsh outfit's owner has denied speculation that the defender is on his way out of the Liberty Stadium, indicating there have been no official offers for the player.
"We have had no phone calls about any of our players," he told reporters. "And we hope that continues.
"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's new Premier League season gets underway on Saturday with a tricky home fixture against reigning champions Manchester United.