The France international is under contract until 2015, but Lyon president Jean-Michel Aulas has reportedly tabled a two-year extension.
But Gourcuff, who was heavily linked with the Gunners this summer, is not thought to be keen to agree the terms which would see his current weekly wage slashed.
A £19m signing from Bordeaux in 2010, Gourcuff has failed to offer a good return on Lyon's investment, and the French club are looking to trim their costs.
As well as Arsenal, the likes of Atletico Madrid and Monaco have been linked with the talented attacking midfielder.
The Le 10 Sport report claims the offer was made to Gourcuff several weeks ago, but it doesn't look like he will be signing it.
That could open the door for Arsenal to return in the January transfer window, with Arsene Wenger thought to be a big fan. But, following the £42.4m deadline day arrival of Mesut Özil, the Gunners boss may feel he doesn't need another attacking midfielder.