Cornet signed for Burnely last summer and impressed for the club on the left hand side, but the Clarets were relegated at the end of the season.
Both Newcastle United and Everton had interest in him, but it is West Ham who have moved swiftly and all but secured his signing.
West Ham have triggered the £17.5m release clause of the winger and have had him in for a medical to secure his signing.
Cornet has completed his medical and West Ham are now ever closer to getting their man in the door ahead of their season starting this weekend.
However, the finalisation of personal terms still remains and the Hammers are in talks with Cornet’s representatives for that to happen.
West Ham will be looking to do that as quickly as possible as they aim to put Cornet at David Moyes’ disposal.
The winger will enhance West Ham’s attacking options and he will be looking to hit the ground running at the London Stadium.