The midfielder has been part of the North London club since 2008, when he joined the youth setup from Stade Lavallois.
Manager Arsene Wenger gave Coquelin his first-team debut just two months after moving to Arsenal, but in recent seasons the midfielder has struggled for regular game time.
In the current campaign, the 26-year-old has started just one Premier League game and has come off the bench in a further six.
According to the Daily Mail, West Ham are leading the chase for the Frenchman's signature, while there has also been interest from his homeland.
The report claims that Arsenal are willing to listen to offers before the transfer window shuts at the end of the month.