Already this window, Chelsea have been linked with West Ham's Andy Carroll, Stoke's Peter Crouch and Burnley's Ashley Barnes in an increasingly desperate search for cover and relief for Alvaro Morata - Fernando Llorente had already chosen to go to Spurs in summer instead.
Antonio Conte's team have also been heavily linked with Roma forward Edin Dzeko, a player proven in the Premier League from his time with Manchester City, but now Giroud is apparently on the ever widening transfer radar as well.
According to the Daily Mirror, Chelsea could try and 'persuade' the Frenchman to make the switch across the capital from north London to west London.
His rumoured move to Borussia Dotmund as part of a possible deal to take Pierre-Emerick Aubameyang to Arsenal crucially looks to be stalling, potentially giving Chelsea the opportunity to swoop in and make an offer of their own.
It will not be a loan, though. Sky Italia's Gianluca Di Marzio has suggested that a loan approach from Stamford Bridge has already been rebuffed, with Arsenal only looking to sell permanently.
Earlier this week, Conte appeared to take aim at his Chelsea superiors over the club's transfer policy. It was heavily reported in summer that the Italian, who guided the team to the title in record breaking fashion in his debut season, was left frustrated by a lack of strengthening.
Conte seemed annoyed after losing to Arsenal in the Carabao Cup semi-final that new signing Ross Barkley, a player who has been injured all season, was his 'only option' on the bench.
He later also commented: "I think the club decide our transfer market. As I said before, my task is to try to improve the team. About the transfer market, from the summer, the club decides every single player... for sure, I don’t have a big impact on the transfer market."