The likes of Sunderland, Aston Villa, Southampton, Premier League new boys Hull City and Scottish champions Celtic are all thought to be monitoring Austin's situation at Turf Moor.
Austin is entering the final year of his contract and the club may be forced to cash-in on their prize asset this summer to bring in funds.
The 23-year-old, who missed the last five games of the season after having appendix surgery, scored 28 goals in 40 games this season.
Burnley now fear Austin may have played his last game for the club with his list of suitors expected to test their resolve to hang onto the hot-shot this summer.
Austin has scored an impressive 82 goals in 155 appearances at Swindon Town and Burnley, and the Championship side will find it hard to turn down a significant offer for the free-scoring forward.