Bournemouth's English midfielder Lewis Cook (L) vies with Everton's James McCarthy (R)
The Toffees swooped to sign McCarthy from Wigan Athletic in 2013, with Roberto Martinez reuniting with the midfielder having worked together at the DW Stadium.
McCarthy, 27, has racked up more than 125 appearances for Everton, but endured an injury-hit campaign last season and fell down the pecking order under Ronald Koeman.
The Republic of Ireland international has played just four times this season having bounced back from injury, and his future at Goodison Park remains a topic of discussion.
The Daily Mail now report that Everton are expecting interest from Crystal Palace in McCarthy's signature this month, but Toffees boss Sam Allardyce wants him to stay.
Allardyce has been linked with a host of midfielders this month, but it now seems that he may give McCarthy a chance to shine in the heart of the Everton engine room.
Allardyce already has Morgan Schneiderlin, Idrissa Gueye, Tom Davies and Beni Baningime at his disposal, but with Muhamed Besic potentially leaving, McCarthy will be needed.
It may subsequently take a huge offer from Palace to tempt Everton into a deal, as Allardyce seemingly values the Irishman more than predecessor Koeman did.