The Gunners are expected to confirm the 25-year-old's arrival this week.
Manager Arsene Wenger intends to play him alongside fellow new signing Lukas Podolski and Dutch striker Robin van Persie in a front three.
Giroud was playing in French's Ligue 2 as recently as 2010, and only made his international debut last November at the age of 25.
The 6ft 3in forward, who was an unused substitute in France’s 1-1 Euro 2012 draw with England scored 21 goals in his club's run to the ligue1 title.
Giroud is one of only two recognised strikers in the French squad, alongside Karim Benzema. He has six caps for his country and one goal.
Earlier this week, Montpellier manager Rene Girard told French radio station RMC that Wenger was leading a chase of top sides for the striker, who was his club's record transfer two years ago when he cost them £1.7m.
" Arsenal are very interested and Giroud wants to play in England, if Arsene agrees with the president, he will sign for them," he said.
Despite speculation about the future of Van Persie, who is out of contract next summer, it is understood Arsenal want to sign Giroud to play alongside him, rather than as a replacement.