The Blues are keen to sign the Swansea City defender this summer.
The Blues are keen to add a central defender this summer, especially if they lose John Stones to Manchester City.
Williams is the man they have identified as an ideal target, with them making their move for the Welsh international over the last week and the player thought to be keen on the switch.
At first it appeared that the deal could be completed swiftly, but the Swans decided to reject Everton’s £10 million bid for his services on Tuesday.
It was reported that Swansea chairman Huw Jenkins was demanding a higher price for his star man before he would sanction a sale.
And according to the Liverpool Echo, Ronald Koeman’s side will do just that and return with an improved bid to ensure they can sign the centre back.
It remains to be seen how high Everton are willing to go however, with them sure to have a limit despite the funds available to them this summer.
With Williams turning 32 next month, they will be keen to ensure they do not pay over the odds for him in what has become a summer transfer window synonymous with large transfer fees.