The Sun reports that the Clarets have contacted the midfielder's advisors as they look to offer him a pathway out of his unhappy stay at Anderlecht.
Defour is unloved by the Anderlecht supporters, stemming from his five-year spell with arch rivals Standard Liege, and is looking to leave the Constant Vanden Stock Stadium this summer according to his agent Paul Stefani.
"Anderlecht's board knows Steven is unhappy, so if an interesting offer comes in we are ready to talk," he said.
"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."