The Wales centre-back has one year left on his contract at the Liberty Stadium, so Swansea might be tempted to cash in on him now rather than risk losing him for nothing next summer.
Williams has been linked with a move to Queens Park Rangers but Sky sources understand that Sunderland boss Gus Poyet has made the first move for him.
Williams, 30, signed a permanent deal with the club at the end of the 2007-08 season after a successful loan spell from Stockport County.
He has been a virtual ever-present rock at the heart of the Swansea defence since, and he signed a new contract in 2012 which keeps him at the club until next summer.
Swansea boss Garry Monk confirmed earlier this week that the club had opened discussions regarding a new deal for Williams.
"We are in negotiations with his (Williams') representatives and trying to put a deal across so that Ash will be here for the future of the club," Monk told the South Wales Evening Post.
"I know Ash wants that, I want that, the club wants that, and that's what we are working towards as we speak."