The Toffees had a £10million rejected by Swansea for their captain, but are readying an improved offer, as they look to clinch the signature of the 31-year-old.
Williams has been a star performer for both club and country, and was crucial in Wales’ run to the semi-finals of Euro 2016, receiving praise for both his personal performances and leadership qualities.
The centre-back has been one of Swansea’s most important players too, and has only missed 8 of the Swans’ 190 Premier League games they’ve played since earning promotion back in 2011.
Everton are keen to push through with a move for the Swans captain, with doubts over the future of John Stones still clouding Goodison Park.
Francesco Guidolin has refused to commit to starting Williams in Swansea’s first game of the new season against Burnley, adding more fuel to the rumours that the Toffees will be returning with a second bid.