The Hornets have approached Lyon over a £20million deal for the Ivorian, according to the Sun.
It is understood that the winger could be tempted by the first-team opportunities and financial lure of a move to England this January, and the Toffees must make sure that they are the side to offer him his chance.
The 22-year-old has struggled to hold down a starting berth for the Ligue 1 side so far this season, appearing just 12 times in all competitions.
He did, however, catch the eye with a brace against Manchester City at the Etihad during the Champions League group stage in September, and is a natural and predatory finisher.
Everton already have options on the flanks, but Theo Walcott is struggling for form and despite the creativity they possess, the Blues don’t score enough goals.
The second half of the season is going to be key for Marco Silva, and could potentially see his side push for Europe, but equally could see him axed if results do not improve.
A positive January could be the difference, and bringing in Cornet could fire his side up the table and towards their continental ambitions.