The Greece Under-21 international has only shown glimpses of his ability since arriving at Turf Moor for £500,000 in the summer of 2001.
Papadopoulos is now entering the final season of his three-year contract, and is attracting admiring glances from Panathinaikos.
Greek sources suggest that Panathinaikos are ready to pay around £250,000 if they cannot obtain the player on a free transfer.
It seems unlikely that The Clarets would accept a total loss on a player that is still so young, although they are currently restructuring their wage bill.
