The Clarets are waiting on a decision from the 28-year-old, with his club Anderlecht prepared to accept their €9m offer.
However, the issue is complicated as the player is thought to prefer a move to United Arab Emirates club Al Jazira, who are only prepared to offer €7m for the Belgium international's services.
Anderlecht were prepared to let the former Porto man leave for €7m two months ago, and it remains to be seen whether Burnley can persuade Defour to make the switch, or Anderlecht cut their losses and accept the offer from Al Jazira.
Defour is looking to leave the Constant Vanden Stock Stadium, as his agent Paul Stefani said: "Anderlecht's board knows Steven is unhappy, so if an interesting offer comes in we are ready to talk.
"He has been insulted and had beer thrown at him by Anderlecht fans, and he did not like that at all. He always gives his best, and thought things would be different.
"A number of clubs have enquired about Steven's situation."
Sean Dyche was reportedly in the stand to watch Defour score in a 5-1 win for Anderlecht against KV Kortrijk on Saturday.