Ronald Koeman's side attempted to bring the French international to the club during the summer but the player elected to remain at the Emirates, citing 'unfinished business' as his motivation.
The 30-year-old is just one goal away from joining the Arsenal 100 club but despite opting to fight fo his place during the summer, has largely been overlooked so far this campaign in favour of Alexandre Lacazette and Danny Welbeck.
Giroud has managed just 100 minutes of action across five Premier League games so far this campaign and is yet to start in the league and should that trend continue he is likely to consider a move away.
English publication 'The Sunday Mirror' reports that the 'Merseysiders' are preparing a huge £40m offer for the striker as they look to add goals to their squad.
The 'Toffees' managed to come from behind to claim three point against Bournemouth on Saturday and end a run of four straight Premier League defeats without scoring but their squad is stil woefully short on firepower.
Is it finally time for Arsenal to cash in on giroud? Or does he still have a role to play for Arsene Wenger's side?