The 28-goal striker was originally available for a knockdown price of £4m, however the Clarets are trying to desist suitors.
They have taken a firm stance on the matter after seeing Jay Rodriguez move to Southampton last summer.
A whole host of clubs have been after Austin, including Aston Villa after impressing scouts.
Jay Rodriguez left Burnley for Southampton last summer
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Burnley co-chairman Mike Garlick refused to get drawn into speculation about Austin's future, telling the Lancashire Telegraph: "He is a Burnley player, he is contracted and we would love to keep him."
Austin has a year left to run on his current Burnley contract.
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