Everton lost against Liverpool at Anfield in the Premier League on Friday.
Everton's Williams has shone on loan for Barnsley, as Middlesbrough's Fletcher did two seasons ago.
Barnsley are reportedly hoping to land Joe Williams from Everton on a permanent basis.
The Everton midfielder could leave permanently this month.
Everton-owned Joe Williams has apologised to the Barnsley fans for the defeat to Leeds United in the Championship on Saturday.
Joe Williams hadn't played a senior game before moving to Barnsley on loan from Everton in the summer.
HITC Sport understands the Everton midfielder is set for Championship loan move.
Joe Williams is held in high regard in the Everton academy.
Manchester United are reportedly pursuing Everton's Joe Williams.