The Swans' skipper was subject to a £10m bid from the Toffees, according to the Liverpool Echo, but that has been swiftly dismissed by the Welsh side.
A club spokesman explained, "We are not looking to sell him and therefore we have turned down the offer currently on the table."
Speculation continues to grow
Transfer rumours surrounding Williams' future with Swansea are certainly something that the Liberty Stadium faithful are becoming used to.
Previously, Arsenal have been linked as the main suitor in search of securing the centre-back's signature, in what is a problem position for them.
Liverpool had been touted as another possible move for the 32-year-old, but now it seems like Goodison Park instead of Anfield is a more likely destination.
Would he fit in at Everton?
Ronald Koeman will be keen to fill the void left by any possible John Stones departure with another centre-half that is comfortable on the ball.
Williams fits the bill perfectly and also comes with plenty of experience, both with Wales and Swansea.
But whether or not he would be prepared to leave, having turned down lucrative advances before, is another matter.
Williams is a club legend
Since signing for Swansea in 2008 from Stockport County, Williams has gone on to amass over 300 appearances for the club; ranging from League One to the Europa League.
He also has two years remaining on his contract, taking him up to 2018 and completing a decade stint in south Wales.
Williams is currently on an extended break following the European Championships, but is expected to be in the squad come their opening game of the season against Burnley.