And Giroud revealed he might have to seek pastures new to cement his place in the France side ahead of the World Cup at the end of the season.
Arsenal have since completed the record signing of Giroud's compatriot Alexandre Lacazette to cast further doubt over his first team chances.
Both the Hammers and Toffees have been busy in the transfer window and reports have linked Slaven Bilic and Ronald Koeman with a keen interest in the 30-year-old, as reported by the likes of the Evening Standard.
But has former Premier League footballer turned pundit Warren Barton just lifted the lid on Giroud's next move?
Barton, who now works as a pundit for USA's Fox Sports, said Giroud's penalty for Arsenal in the Community Shield win over Chelsea was his 'last kick for the club before a move to West Ham'.
Whether Barton has any inside information on a potential deal it remains to be seen.
But Hammers fans will welcome the news as they bid to find a partner for Javier Hernandez.
It would be bad news for Toffees boss Koeman who has been hoping to continue his own impressive summer business by replacing Romelu Lukaku with a proven Premier League striker in the shape of Giroud.
If Bilic does land Giroud it would be a measure of revenge over the Merseysiders after they stole Nigerian forward Henry Onyekuru from under West Ham's noses earlier this summer.