Cornet was not able to prevent Burnley being relegated to the Championship last season, but his performances at Turf Moor will likely be enough to keep him from playing in the second tier.
He has substantial interest, with Everton and Nottingham Forest two of the teams keen to offer Cornet a new home.
However, it is suggested that Cornet would like to join a team that can offer him European football next season, something Everton and Nottingham Forest cannot do.
Cornet has a release clause in his Burnley contract, set at £17.5m, but no club are yet to trigger it.
The 25-year-old winger has played in Europe before, turning out in the Champions League and Europa League, and is keen to experience it again.
He clocked 26 appearances for Burnley in the Premier League last season and chipped in with nine goals.
Cornet played a key role in Burnley’s win over Everton in April, scoring and assisting for the Clarets.