Relegation-threatened Livorno have sacked Domenico Di Carlo after just 88 days in charge and re-appointed Davide Nicola.
Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
Aston Villa and Newcastle are both reported to be chasing Austin, who is the top goalscorer in the country with 15 goals this term.
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 insists they have not had any approaches for Austin and that they want to keep hold of the prolific striker.
"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 said. "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."