Current caretaker manager Steve Davis has already thrown his hat into the ring following the departure of former boss Steve Cotterill at the beginning of the week.
However, Blackpool boss Simon Grayson has been strongly linked with the Turf Moor hot-seat.
"Burnley Football Club can confirm that a shortlist has been formulated for the vacant manager's position at Turf Moor," said a club statement.
"Five men remain in the frame to succeed Steve Cotterill and the club hopes to make a formal announcement by the middle of next week."
Earlier in the week, the Clarets were denied the opportunity to speak to Sheffield Wednesday boss Brian Laws by the Hillsborough board.