The Gunners made their move for the 25-year-old prior to the tournament but boss Arsene Wenger has been made to wait until he has fulfilled his international commitments before wrapping up the transfer.
Giroud is now expected to have a medical at the Emirates Stadium with Les Bleus now out heading home after being dumped out by defending champions Spain at the weekend.
And the striker spoke of his excitement at moving to Arsenal and linking up with some of his France teammates in North London.
“I am so happy to be signing for Arsenal – it is something that represents so much for me,” Giroud is quoted by the Sun.
“I’ve always dreamed of playing in the Premier League and Arsenal are a great club with many French players.”
Meanwhile, Bordeaux president Jean-Louis Triaud has denied reports that the club were looking to bring Marouane Chamakh back to the Stade Chaban-Delmas this summer.
The Moroccan striker joined Arsenal from the Ligue 1 side on a free transfer two-years ago but has struggled to replicate the goalscoring form he displayed during his time at Les Girondins.
With the imminent arrival of Giroud and acquisition of Lukas Podolski the 28-year-old is surplus to requirements but Triaud believes Chamakh will remain in England after turning down an approach back in January.
“It is not worth making things up. I tried to contact him six months ago when the transfer window was open,” Triaud told Sport 10.
“Since I crossed paths with him at the Parc des Princes, he was able to tell me he was very happy in London, loved Arsenal and was fine where he was.
“We’re talking about things which do not exist.”