The 27-year-old, who has scored three goals in 18 appearances this season, could be an interim option for Arsène Wenger ahead of a summer move for Schalke's Julian Draxler.
Draxler's contract includes a £37 million buy-out clause, and the Gunners seem reluctant to sanction another hefty outlay following the £42.5m arrival of Mesut Özil in September.
Arsenal tried to sign Gourcuff on loan last season and Wenger, who has admired the attacking midfielder for years, has contacted Lyon, say the Times.
Once billed as the "new Zinedine Zidane", Gourcuff cost £18 million when he moved to Lyon from Bordeaux in 2010 but has struggled to justify the hype he initially generated in France.
Gourcuff joined AC Milan in 2006 but returned to his homeland after a disappointing couple of years in Italy.