McArthur joined Crystal Palace from Wigan Athletic in 2014, and has since gone on to make 95 appearances, scoring nine goals. It is thought that the Scotland international might not be in the plans of new manager Frank de Boer.
Burnley's coach Sean Dyche is an admirer and wants McArthur to join his side, with Aston Villa also interested in his services.
Aston Villa manager Steve Bruce is working on a deal for Stoke City's Glenn Whelan but he also wants McArthur.
Burnley are favourites to sign McArthur, although they need to sell more players before they can make a serious bid.