According to The Athletic, the two clubs are in talks for the player who has also previously garnered interest from Newcastle and West Ham.
Cornet only joined Burnley last summer for £13.5million, scoring nine goals in 26 league appearances, but was unable to save the Lancashire side from dropping into the Championship.
At his previous club Lyon, the Ivory Coast international scored 51 and made 29 assists in 252 games, helping the Ligue 1 club to the semi-finals of the Champions League in 2020.
Given that he is a player capable of playing European football, his move to Burnley was always seen as something of a coup and as a result it is highly unlikely that he will stay at Turf Moor to play in the second tier.
It is also reported that the 25-year-old has a relegation release clause of just £17.5m which should make him more appealing to potential buyers.
He could become Everton’s second signing of the summer, after Frank Lampard snapped up former Burnley centre-back James Tarkowski on a free.
Burnley are currently in the process of rebuilding their squad under new manager Vincent Kompany, with several other notable names already gone, including Nathan Collins to Wolves, Nick Pope to Newcastle and Wout Weghorst to Besiktas on loan, while club captain Ben Mee has also been released.
Everton are also set for a pivotal summer of transfers having narrowly avoided the drop last season, with star forward Richarlison sold to Tottenham for £52m as the Toffees look to balance the books after years of overspending.
Cornet would seem to be an ideal and relatively inexpensive replacement then and will add some much-needed versatility to Lampard’s squad, as he can play as a left-winger, striker and left-back.
For more stories like this, check our sport page.
Follow Metro Sport for the latest news on
Facebook, Twitter and Instagram.