The Frenchman was high on the Toffees radar during the summer, but Arsenal refused to do a deal, while the striker also committed himself to the Gunners, saying he wanted to fight for his first-team place.
Giroud could have a change of heart though as he again struggles to force his way into Arsene Wenger's side, and with the World Cup fast approaching, he knows he needs to be playing regular first-team football to make the squad.
Everton could offer him that and the chance to make the switch to Goodison Park when the transfer window opens in the new year will be a tempting one, if Arsenal back down and agree to cash in.
Blues boss Ronald Koeman has yet to replace Romelu Lukaku, who joined Manchester United, with an out-and-out goalscorer and reports are claiming he is willing to splash out as much as £40 million to land his man.
Everton are not the only interested club, with West Ham United also said to be keen.