The 30-year-old was a target for Lyon, West Ham United and Everton after making clear at the end of last season that he was not happy with his lack of minutes.
Giroud, handed just 14 Premier League and Champions League starts combined in 2016-17, was then pushed further down the pecking order following the arrival of Alexandre Lacazette.
Opening up on his decision to remain in place, despite the interest from elsewhere, the Frenchman told reporters: "It is a very private decision. It is true that I was close to leaving the club because I wanted to play.
"After big reflection with my family, my close friends, I decided to stay. I thought the story wasn't finished at Arsenal. The club gave me a lot and I still want to win some more trophies with the club."
Giroud, a £10m signing from Montpellier in 2012, has featured four times from the bench so far in the Premier League this term.