The Toffees, who could potentially be without John Stones as Manchester City's interest in the 22-year-old Englishman has continued to intensify in recent weeks, are aiming to sign adequate replacements.
Everton eyeing adequate defensive replacements
Amongst them is Williams, with the 31-year-old having flourished with his native Wales at this summer's European Championships as they progressed to the semi-final stage before being defeated by eventual winners Portugal.
Swans will be eager to keep their skipper
Williams has played for the Swans since 2008 and although the captain is a key man for Francesco Guidolin's side, the Italian himself has admitted he's unclear about what the future holds for Williams.
If the report is valid, Swansea could ask for slightly more than the £10m that has been widely reported, especially considering he's notched up over 300 club appearances and has adequate Premier League experience - which could prove beneficial for a side that has struggled for defensive solidity in recent seasons despite their quality across all areas of the pitch.
Toffees aiming for tougher backline
Everton, who conceded 55 goals last season, will be aiming for better defensive displays on the whole in the forthcoming campaign - especially under new management with Ronald Koeman, formerly of Southampton as the club's new boss.
They've also been linked with the likes of Porto's Bruno Martins Indi, Sunderland's Lamina Kone and Anderlecht's Kara Mbodji as the new season draws ever closer, and they're expected to complete a lot of transfer business this upcoming week with the futures of Axel Witsel, Romelu Lukaku and a few other players set to be finalised.