The report claims that he is set to undergo a medical at Stoke City on Thursday after the two clubs agreed a loan arrangement.
At 33, he is approaching the twilight of his career and it appears the former Swansea City defender may have already played his last game for the Toffees.
Sky Sports claim that Williams is close to joining Gary Rowett’s side where he will be reunited with his national teammate Joe Allen.
Williams has one year left on his contract and so Everton are likely to miss out on getting a permanent fee for the experienced defender.
His departure will vacate the space for a potential new arrival with Everton looking to sign a top centre-back this summer.