Reports in France are suggesting Lyon are desperate to trim their wage bill, and are struggling to convince Gourcuff – incidentally the clubs highest earner – to sign a new deal.
The France international still has two years to run on his current deal, but has been left unimpressed by an offer of an extension from club president Jean-Michel Aulas which would see his current weekly wage slashed.
The 27-year-old is thought to earn around £90,000 a week at present, but may now seek a January move as he is not happy with the clubs new offer.
The news has alerted Arsene Wenger, who has been tracking his fellow countryman for some time.
The Gunners were linked with him heavily in the summer, with reports in August suggesting Gourcuff was willing to take a pay cut to force through a move to the Emirates.
Wenger has been told he has money to spend in January, although he has made signing a centre back and a centre forward his main priorities.
Lyon chairman Jean-Michel Aulas has admitted Gourcuff could be on his way, but has confirmed he will only sell his prized asset to a foreign club.
“All that I know, if I understood correctly, is that he was ready to lower his salary to go to Arsenal, but that’s not the same thing,” he said.