Giroud, who is part of Laurent Blanc's Euro 2012 squad, is thought to command a transfer fee in the region of £13million. The 25-year-old scored 25 goals for his club last season to help Montpellier win the Ligue 1 title, igniting the interest of several top clubs.
However, it appears Arsenal are set to secure the promising Frenchman's services, having already signed Germany marksman Lukas Podolski.
Montpellier's deputy president Laurent Nicollin told French newspaper Midi Libre: "Discussions are happening and going well, as part of the (release) clause set. Things are moving serenely."
Once the deal for Giroud is complete, Arsenal could then look to offload some of their fringe players, including Denmark striker Nicklas Bendtner and Morocco international Marouane Chamakh.