The Blues are keen on the Wales international but Huw Jenkins wants a better price.
Wales' Ashley Williams and teammates celebrate at full time
The Blues are expected to be busy improving their squad this summer, with Ronald Koeman given a big transfer fund by Farhad Moshiri.
He has made just two signings so far, with Maarten Stekelenburg and Idrissa Gueye arriving at Goodison Park.
Swansea's Ashley Williams celebrates scoring their first goal
One of his aims is thought to be improving the defence, with that area of the team struggling under Roberto Martinez last season.
He appears to have identified Williams as the man he wants, with Wales Online reporting that they had made a £10 million bid for his services.
However, Swansea City have dug in their heels over their star man, with the Mirror reporting that they have rejected that offer.
They report that chairman Huw Jenkins is demanding an improved offer for the Wales international.
Ashley Williams celebrates scoring the second goal for Swansea City
It remains to be seen whether the Blues will return with such a bid or turn their attention elsewhere.
There is no denying that Williams would be a smart buy, but Ronald Koeman may be concerned about paying a large sum for a player who turns 32 this month.
Everton manager Ronald Koeman