It has been 12 years since Giroud last played in the French top-flight, having left Montpellier to join Arsenal after winning the Ligue 1 title in 2011-12.
Lille president Olivier Letang has already confirmed the club's interest in him, saying: "I’d love to see him join us. He brings everything we seek, tactical fit, experience, high standards, and great human values. But he belongs to another club, and I don’t see how it could happen easily."