Cornet is keen to move on from Burnley after they were relegated from the Premier League last season and his stock is high in the top flight following making an impression at Turf Moor.
Everton have shown an interest in signing him, but Burnley have rejected their offer of a loan with the option to buy and will only accept bids to sell.
David Moyes has the cash to spend despite forking out big fees for Gianluca Scamacca and Nayef Aguerd, and the West Ham boss wants Cornet.
It has been claimed that the Hammers are prepared to offer up to £20m to Burnley for the winger this summer.
The West Ham boss is aware that the club will have to top other offers and is keen on getting a deal done quickly.
The two clubs are in talks as West Ham look to assess what kind of bid would do the trick for Cornet.
The winger had a fixed fee in his contract if Burnley were relegated but that became public knowledge, which annoyed the Clarets.
They now want more than the figure in his contract and it seems West Ham are prepared to play ball.
