The Toffees have had an initial offer of £10million rejected by Swansea City but the Daily Mirror is of the belief that an improved bid could be forthcoming.
Williams, 31, is one of Koeman's summer priorities as he believes his leadership and ball-playing abilities will have an enormous impact at Goodison Park.
Wales captain Williams has been with the Swans since 2008 and has never shown any desire to leave the Liberty Stadium but it is understood that he is intrigued by Everton's interest.
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