The Gunners appear to be closing in on a club-record deal for Lyon striker Alexandre Lacazette.
And that has further raised doubts over Giroud's long-term future at the Emirates.
The France international, 30, struggled for regular game last season, starting just 11 Premier League matches.
He was already being linked with a move away from north London amid interest from West Ham and Marseille.
And, according to The Sun, Everton are now ready to enter the race for his signature.
They claim Giroud could become a priority for Ronald Koeman if Romelu Lukaku leaves Goodison Park this summer.
The big Belgian is being heavily linked with a return to Chelsea.
And The Sun say Koeman views Giroud as an ideal replacement for his star forward.
The former Montpellier man has two years left to run on his Arsenal contract.
But The Sun suggest he would be willing to leave the Gunners in search of a regular starting spot.
The Toffees have already splashed out a combined fee of £53m to sign goalkeeper Jordan Pickford and midfielder Davy Klassen this summer.
