New Toffees manager Ronald Koeman has made the 31-year-old defender his third summer purchase after Senegal midfielder Idrissa Gueye and goalkeeper Maarten Stekelenburg.
The Merseyside club are also said to be in talks with Crystal Palace midfielder Yannick Bolasie and Sunderland defender Lamine Kone.
The capture of Williams comes a day after John Stones left Goodison Park to join Manchester City for £47.5m.
Williams joined the Welsh club in 2008 from Stockport for a fee of just £300,000 and went on to make over 300 appearances for Swansea.
He was also a major figure in Wales' incredible Euro 2016 run that saw them make the semi-finals of the competition.