The Welshman proved heroic at EURO 2016 as his team made it to the semi-finals and has been a stalwart at the Liberty Stadium amid their stunning rise to the Premier League.
He was absent from the Swans’ pre-season against Wolves and is reported to have played his last game for the Jack Army.
The Toffees are in dire need of reinforcements and could yet lose John Stones to Manchester City, with just over two weeks until the start of the season.
Though Koeman has lavish funds to spend at Goodison Park, recruitment has proved difficult since moving to Merseyside from Southampton, missing out on Stoke City’s Marko Arnautovic.
Williams is expected to double his £2m-per-year wages should he make the move to Everton after establishing himself as a legend in South Wales.