"We have made enquiries about him and it is part of our plan to strengthen the squad," said Burnley Chairman Barry Kilby.
"It is one of the many avenues we are looking to try and make sure we go forward.
"Stan Ternent (Burnley manager) has had a conversation with Walter Smith about his availability so the ball is in Everton's court.
"But it has to be sorted out financially and the move has to be right for both parties. So the decision will ultimately be down to Paul Gascoigne," Kilby told the club's official website.