Bolton plan bid for Middlesbrough defender Williams
Coyle failed with a summer bid for the £3million-rated star, who is yet to agree an extension with 18 months left on his current contract.
Williams could be lured away if promotion-chasers Boro fail to reach the Premier League this season.
Coyle, who took Williams on loan at Burnley, is set to test the Teessiders' resolve with a fresh move in January.
Williams, 23, on international duty with Australia, said: "I'm happy to sign with Boro if the deal is right but, just now, it's not. Until it is, I'll sit on what I have.
"I've tried hard for the last three years to help get Boro back up - hopefully this season will be the one. If not, I will have a serious look around."