Fekir came close to joining Liverpool earlier in the transfer window as the Reds entered talks over a £60 million (€68m/$80m) transfer with Lyon. The attacking midfielder even had a medical with the Reds as a deal looked imminent, only to see it fall apart before the World Cup .
The collapse was quickly followed by rumours Fekir could end up heading to Real Madrid, but Lyon boss Bruno Genesio and president Jean-Michel Aulas have since suggested he could stay in Ligue 1 for another year .
After captaining Lyon to third place in Ligue 1 and scoring 18 goals in the top flight last term, speculation continues to build around Fekir, and Giroud has suggested he and Chelsea could be a match made in heaven.
"I spoke to him when we were with the France team and his transfer to Liverpool was aborted," Giroud told reporters.
"After that, it was a question about Chelsea. Even Eden had a couple of words with him.
"We'd welcome him, he's a great player and of course we'd welcome him with pleasure. I don't know if it's news but the market finishes on Thursday."
Chelsea have made only one major transfer this summer in the shape of Jorginho, who joined from Napoli, but coach Maurizio Sarri confirmed last week that he hopes to land another midfielder before the transfer window closes.
Fekir is not the only option, however, as the London side are in talks to land Mateo Kovacic from Real Madrid .