Cornet is keen to move on from Burnley after they were relegated from the Premier League at the end of last season.
Several Premier League clubs have shown an interest in him, with Newcastle, who are keen for a wide-attacker, having tried to strike a deal with Burnley for his signature in recent days.
However, West Ham are also keen on Cornet and they have held talks with Burnley about taking him to the capital.
But it has been claimed that the Hammers want to sign the winger on an initial loan.
Talks are ongoing but West Ham are keen to avoid committing to paying a transfer fee to sign Cornet.
It is unclear if such a formula would be acceptable for Burnley, especially given other clubs are also keen to land the former Lyon man.
Cornet has a £17.5m release clause following Burnley’s relegation and Everton have also spoken to the Clarets about him.