Cornet is ready to leave the Clarets following last season's relegation.
Cornet's deal carries a £17.5m buyout clause.
The Daily Mail says Fulham are now making their move.
The Cottagers missed out on a deal for Italy international Federico Bernadeschi last week when he opted to join Toronto instead and have now gone back for Cornet who was on their radar at the start of the window.
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