The Gunners have been increasingly linked with the 24-year-old in recent months as they consider possible replacements for Alexis Sanchez and Mesut Ozil, both of whom are out of contract at the end of the season.
Should Arsenal manage to push through a deal in the January transfer window, they could allow both Sanchez and Ozil to also leave in order to avoid losing them on free transfers during the summer.
The Daily Star reports that the North Londoners are now in advanced talks with Lyon, who also provided Arsenal's current most expensive player during the summer when Alexandre Lacazette moved to the Emirates Stadium.
Arsene Wenger is now willing to surpass that £52m deal for Fekir, with the move potentially being funded by the £30m departures of Sanchez and Ozil.
Fekir - who has also been linked with Barcelona - has directly contributed to 17 goals in as many games across all competitions for Lyon this season, scoring 13 times and assisting four more.