Everton's bid of £10m was turned down by the Swans, who are not interested in selling their captain and one of Wales' Euro 2016 heroes.
However, it is thought that the south Wales club may be tempted with a £15m offer, but it remains to be seen whether Everton would be prepared to pay this much given that Williams is 31 years old.
Everton's intentions to sign a defender could mean that John Stones is on his way to Manchester City. Despite this possibility, there has yet to be talk of any agreement in place.