The 21-year-old has been monitored by Dean Smith's side throughout this window, as they are looking to strengthen their attacking options.
However, according to Football Insider, the Toffees have now expressed their interest in McNeil and they are "going strong" to try and complete a deal this summer.
It is understood, however, that Everton would need to sell players first to generate funds before they can consider entering negotiations.
McNeil's current contract runs until June 2024 and Burnley are under no pressure to sell their star winger.
The Englishman, who scored two goals and provided five assists in 36 Premier League appearances last season, would be open to a move away from Turf Moor to play at a higher level. body check tags ::