Maurizio Sarri is eager to bring Higuain to Chelsea before the transfer window closes on August 9 and talks have been ongoing over a deal for the Juventus striker.
Giroud, who only joined the Blues from Arsenal in January, has been earmarked as a potential player that could leave Stamford Bridge.
Alvaro Morata’s long-term future with the west London outfit appears to be under scrutiny after an indifferent campaign last term.
And according to the Daily Mirror, Marseille have enquired about signing Giroud, who they tried to sign last summer.
The report suggests the Ligue 1 outfit have resurrected their interest after failing with an approach to sign Mario Balotelli.
Sarri is keen to stamp his mark on the Blues and he has so far signed two players – Jorginho from Napoli and Rob Green on a free transfer.
Giroud on the other hand currently has 12 months left to run on his existing deal.
Meanwhile, ESPN pundit Paul Marriner reckons Morata will stay at Chelsea and force out Giroud.
“In his younger days I would have said yes Giroud would sort of fit,” Marriner said.
“But now it looks like Morata may stay, I always put Morata in front of Giroud because I think Morata got a bit of a rough deal last year. He had a back injury, he wasn’t 100 per cent fit.
“So maybe he’s (Sarri) seen something in pre-season that he likes about Morata.”