The Daily Mail says the Toffees will rival West Ham for Burnley forward Cornet after the Clarets dropped into the Championship.
The 25-year-old is understood to have a release fee of £17.5m, and - unlike his current club - looks set to remain in the Premier League.
Elsewhere, Blues boss Frank Lampard fancies a move for Brentford ace Christian Eriksen, but faces competition from the Dane's former club Tottenham.
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