The Wales international is about to enter the final year of his contract at the Liberty Stadium, and is understood to have indicated that he is ready to move on.
The Swans are desperate to keep the centre-back by offering him a new deal, yet the29-year-old feels that now is the time to make a big move.
Arsenal have been linked with Williams on a number of occasions in the past few years, but each time a swoop has been unsuccessful with the South Wales side having indicated that they will not sell their man.
But now a £3m offer may be enough, with Swansea unlikely to be willing to let their on-field hero walk away for free in 12 months’ time.
Liverpool have also been linked with the experienced defender, however their interest is thought to have cooled in recent times.
As well as Arsenal, foreign sides Napoli and Lille may look to make moves.