Austin is the top goalscorer in the country with 15 goals this term, and having already tried to sign back Andy Carroll in the summer, Alan Pardew is obviously looking for another striker.
Austin is in the final 18 months of his contract at Turf Moor, but the club have yet to open talks over a new deal for the former Swindon hot-shot.
Burnley co-chairman John Banaszkiewicz: "Charlie's come a long way, he's a big asset for us and he has helped us secure some vital points this season.
"He is on a long-term contract, expiring in 18 months, and he has really blossomed.
"We've had no approaches from any clubs. We would very much like to keep him at Burnley. He seems to be very focused on scoring goals for us."