Blues boss Ronald Koeman is desperate to land a new striker when the transfer window re-opens in the new year and Giroud remains on his radar.
The Frenchman came close to joining the Toffees in the summer but turned down the move in favour of staying with the Gunners, and this week revealed he made the right choice as Koeman's men have struggled so far this season.
However Giroud now has other options as both Juventus and Inter Milan are in the hunt for a new striker and reports in Italy claim that the 30-year-old is a target.
Giroud is struggling for regular action with Arsenal and he could be tempted by a move to either club if he receives guarantees about the amount of game time he will get ahead of the World Cup next summer.
The striker has just helped France secure qualification to the finals in Russia and will want regular football between now and then to ensure he wins a place in the squad, something he would get at Everton if he had a change of heart.