Reports from The Sun claim the 25-year-old Ivorian international will sign a five-year deal with the East London club, with the inclusion of a one-year option, which could take him up to 2028.
Unless the paperwork is completed by midday he won't be registered in time to feature in Sunday's opening fixture against Manchester City.
However, it is highly unlikely West Ham manager David Moyes would want to throw the left-sided wing back straight into a big match with the Premier League champions, especially without training with the rest of the first team.
Other new signings Gianluca Scamacca and Flynn Downes could make their debuts on Sunday, though new defender Nayef Aguerd is expected to miss out through injury.
When questioned about Cornet's move Burnley manager Vincent Kompany ruled his player out of this weekend's match with Luton.
"At this point… my focus is on incoming players or players who are fit or available for the weekend," he said.
"At this moment in time Max won't be involved on Saturday, we'll manage that week by week."