Arsenal said the player will remain in London “against our clear instructions” and added they are “very disappointed” by his actions in a short statement on the website.
Following Arsenal’s 3-0 win against Colorado Rapids, Unai Emery said: “I spoke with him three times for his situation. We wanted him to come with us but he decided to stay. For me now, it’s clear. We need to keep moving ahead. We’re playing with the spirit to create a new way.”
Giroud, who spent six years at Arsenal before moving to Chelsea in January 2017, quickly leapt to Koscielny’s defence.
The French forward said the Arsenal defender has always been “grateful” and “respectful” and that he feels “sad” for him because of the image the media has given him.
"Lolo is a good friend of mine," said Giroud in an interview with BBC Sport. "I've known him now for 10, 11 years. We played together in the second division [in France, at Tours].
"He's been playing for Arsenal for nine years; he's been the skipper for so many years - very respectful, very professional, always the best example for the young players.
"The club gave him the opportunity to play at Arsenal, and it was a big, big opportunity for him to show his quality in the Premier League. I think he has always been grateful about that and respectful to the club. I don't understand how we have got to this situation.
"I feel very sad for him because he's a great person and the media wants to give a bad image of him.
"I'm not at the club, I don't know what really happened - but one thing for sure is Lolo is very hurt by what's happening and he's an emotional person."
Giroud also opened up on his departure from the club after scoring over 100 goals in all competitions during a spell where he won three FA Cups.
"I didn't want to leave," Giroud said. "I spoke with the boss - at this time, Arsene [Wenger] - and I told him: 'Look, gaffer, I need some game time to play at the World Cup, that's massive for me.'
"At the end of the day, we won it, so it was very important for me to leave.
"It was tough for me, but it was amazing to sign for a club like Chelsea. It was like an improvement for me because - OK, it's Manchester City now - but Chelsea have been the best club in England over the past 10 years.
Giroud’s relationship with the Arsenal supporters was turbulent throughout his spell at The Emirates, although he expressed his sympathy for the disgruntled fans that have recently launched a petition to change the way the club is run.
Almost 100,000 supporters have signed a petition promoted by a number of supporter groups in an attempt to change the way the club is run under the banner #WeCareDoYou.
"I can understand the fans," said Giroud. "With Arsenal it was difficult to fight for the title. We were always fighting for the Champions League places," he said. "I won a few FA Cups and Community Shields. When you can't win the league, winning the FA Cup is quite nice, so I've got amazing memories.
"Chelsea is a club where you have maybe more expectations, higher expectations. They were winning more titles these past 10 years and obviously Arsenal wants to come back to the 'Invincibles' days, but it's not easy. I'm not going to say we are doing things better at Chelsea. We just try to be as competitive as we can, to step up and win more trophies."
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