The Yorkshire club are believed to have offered £1m to turn the midfielder's current season-long loan into a full-time deal but the Blues have rejected the bid.
Williams, who was a member of the Everton U23s that won Premier League 2 last season, was on the fringes of the first team towards the end of last season and it is felt he could have a big future at Everton.
Tykes boss Paul Heggingbottom is quoted saying: "We did [make a bid], that's come out. I don't know where it's come from, whether it's been Everton's end, whether it's been agents.
"Yeah, there was a bid but that's it. [It] was rejected. I don't know [if we will make another offer]."
Williams has made 28 appearances for Barnsley and wowed their fans with a spectacular 30-yard goal against Burton Albion earlier this season that was named Championship Goal of the Month in October.