Everton had sent the 20-year-old out on loan to the Belgian champions for the season, but their attacking struggles have led to Ronald Koeman's side considering recalling the youngster.
Arsenal on the other hand are in fifth and will be looking for a replacement for star winger Alexis Sanchez who is likely to leave the club in the January window.
The club-record signing of Giroud's France team-mate Alexandre Lacazette left his future up in the air.
"I was very close to signing for another English club, Everton, but I made the best choice I think", Giroud told France's Canal Football Club. As per the provided Sun link, Everton would be reluctant to sell him but an offer worth 15 million pounds ($20million) could be very tempting for the Merseysiders.More news: Nadal eases past Kyrgios to claim China crown
Giroud has seen his game time diminish this season.
"I also have a lot of respect for Lyon, who were also interested at one time". Everton have started the season dismally despite breaking the bank over the summer with new signing Gylfi Sigurdsson, only winning twice in seven games and sitting at 16th place in the Premier League. "But a return to France is possible one day".
The France worldwide has yet to start a Premier League game for Arsenal this season with only a handful of first-team appearances in cup competitions.
With Giroud hoping to make France's squad for the 2018 World Cup, he will have to make a bigger impact as the campaign progresses.