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Kee has attracted the attention of the wealthy West Sussex club after scoring 12 goals since his summer switch to the Pirelli Stadium from Torquay.
Albion boss Paul Peschisolido told the club's official website: "Crawley have made an approach for Billy Kee with a substantial offer.
"We have spoken to Billy and he feels it would not be the right move for him at this time."
Crawley, promoted to the Football League as Conference champions last season, are currently third in League Two and reached the fifth-round of the FA Cup on Saturday with a 1-0 win at Championship club Hull.
Burton are ninth in the table and just a point outside the play-off places.
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