The Daily Mail says Wenger is a long term admirer of the 27-year-old attacking midfielder who has scored three goals in 13 league appearances this season.
Lyon have been hit by financial problems in recent years and despite the club president stating no-one is for sale the Gunners inquired about their asking price at the start of January.
Gourcuff is an alternative to Schalke's German star Julian Draxler who would cost considerably more, and is still injured for the next few weeks.
Wenger and the Arsenal board are due to meet this week to discuss Draxler but Gourcuff could be an easier deal to get over the line before the transfer window closes on Friday.
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