Giroud, part of Laurent Blanc's Euro 2012 squad which was knocked out by Spain in the quarter-finals on Saturday, 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.
Last week, Gunners boss Arsene Wenger revealed a deal was "90%" complete as he looked to add the promising Frenchman to his squad, which has already been bolstered by the arrival of Germany marksman Lukas Podolski.
Speaking to French newspaper Midi Libre, Montpellier president Nicollin confirmed the transfer had now been completed after Arsenal met the player's release clause.
Nicollin said: "I wish him well."
Arsenal have not yet made any official comment, but Press Association Sport understands an announcement by the Barclays Premier League club is imminent.
Once the transfer for Giroud is formally complete, Arsenal could then look to offload some of their fringe players, including Denmark striker Nicklas Bendtner and Morocco international Marouane Chamakh as well as Mexico forward Carlos Vela and Brazilian midfielder Denilson.
However, Nicollin maintained a move to take Chamakh in the opposite direction back to France was not part of the Giroud deal.
Wenger, who is at the European Championship working as a media pundit, is convinced Giroud can be a big hit at Arsenal.
"I think he has exceptional potential and will integrate very well in our collective and our way of playing," he said last week.
"He is a real team player and knows how to combine with other players and I'm sure he will do well with us."Bendtner spent last season on loan at Sunderland.
The 24-year-old played for Denmark at Euro 2012, scoring twice in the 3-2 defeat by Portugal.
The player's agent maintains there is "global" interest in his client.
Speaking to bold.dk, the website of a Danish newspaper, Bendtner's representative Jesper Lynghus said: "The interest is overwhelming, just as it should be after the way Nicklas performed at Euro 2012.
"We are getting closer day by day.
"We are not commenting on individual deals at the moment, but it is fair to say the interest in Nicklas is global."
Fulham, meanwhile, are reported to be ready to step up their interest in a £5million move for Arsenal midfielder Aaron Ramsey following the departure of their captain Danny Murphy to Blackburn.