The Swansea captain has been at the Welsh club since 2008, and lifted their first ever piece of silverware alongside Garry Monk in 2013.
But Wales Online are reporting that Williams is keen on a move to join Ronald Koeman at Everton, and that a £10million switch will be completed within days, leaving Swansea without their captain less than 2 weeks before the new season starts.
Francesco Gudiolin was keen to keep hold of Williams, after the Italian himself committed to the club this summer to stay on as permanent manager. But the 32-year-old, who has been repeatedly linked with a move away from the Liberty Stadium over the past few seasons, finally looks set to depart.
Williams joined for £300,000 in 2008, and has since gone on to make over 350 appearances for the club.
But with rumours of John Stones getting closer to a switch to Manchester City, Everton have already got his replacement lined up, and are finalising the deal now.