The 33-year-old has made just four Premier League appearances for Frank Lampard's side totalling 120 minutes, with Tammy Abraham and Michy Batshuayi preferred to the former Arsenal hitman.
Now Giroud, who helped France to a second World Cup win last year, could be on his way back to his homeland if Lyon decide to pursue the 94-cap striker.
His contract at Stamford Bridge is up in June 2020 and Les Gones could make a move in the January transfer window. It is believed the Ligue 1 side view Giroud as competition for Memphis Depay in attack.
Giroud is understood to be open to a move away from Chelsea as he seeks a starting slot in the France squad ahead of next summer's Euro 2020 tournament.
The former Montpellier striker said: "I think I have a good few years in front of me. I'm 33-years-old but I still have the legs - physically I'm capable of playing on for a few more years.
"I have a good lifestyle, I want to have fun and play as many matches as possible. My priority is to stay at Chelsea but if I'm forced to make a choice, like when I left Arsenal, I'll do it."
Both Manchester United and Juventus are leading the chase for former Hoops striker Dembele, who joined Lyon for £19.7 million in the summer of 2018. The 23-year-old has hit seven goals in ten games so far this campaign, giving him a total of 27 in 59 for Rudi Garcia's side, but he is in high demand as a result.
Ole Gunnar Solskjaer is understood to be a fan of the Frenchman and has been linked with a move in January which could set the wheels in motion for Giroud's return to Ligue 1.
Old Trafford scouts ran the rule over the former Paris Saint-Germain youngster in Lyon's matches against Brest and Zenit St Petersburg in September.
Assuming Dembele goes for a large transfer fee - earlier this year he was valued at £40 million by Lyon but that figure could now have risen - Celtic are in line for a sizeable chunk of any deal.
The Hoops are believed to have a sell-on clause entitling them to as much as 15 per cent of any potential deal. With Dembele under contract at Lyon until the summer of 2023, the Scottish champions could net a small fortune if their former striker is on the move in January.