The defender is expected to complete a move from Swansea City imminently.
The Blues are in the hunt for a new central defender after John Stones officially joined Manchester City on Tuesday.
Williams appears to be top of their wish list, with them already having seen a bid rejected by Swansea City for their captain.
However, they have returned with an improved bid after Williams made it clear to chairman Huw Jenkins that he would like to leave.
According to the report, the Wales international is thought to be on his way to Goodison Park having already said his farewells at the Liberty Stadium.
The move will bring about an end to an eight-year spell with the Swans, during which time he made over 300 appearances and oversaw two promotions, a League Cup win and a short stint in Europe.
The deal will come as great news to Everton boss Ronald Koeman, who has so far been frustrated in his attempts to bring in players this summer.