The Republic of Ireland international is in the final year of his contract at Goodison Park and has made just one Premier League appearance since breaking his leg in January 2018.
Sky Sports News reports that Palace have come in with a formal bid for the 28-year-old.
West Brom failed with an effort to sign the player on loan in January, according to the outlet.
McCarthy returned to the Toffees’ matchday squad at the start of the year but failed to break into Marco Silva’s side, earning just one substitute appearance against Manchester United before the end of the campaign.
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