Toffees boss Marco Silva is on the lookout for a new striker and it has been suggested he could make a move for Giroud if Chelsea are open to the idea of cashing in on the Frenchman in the new year.
The 32-year-old is out of contract at the end of the season and there has already been talk of West Ham making a move as they look to solve their issues in front of goal, and Everton could join them.
Giroud has played a key role for Chelsea this season when battling with Alvaro Morata for a place in the starting line-up, and came off the bench in Saturday's 2-2 draw against Manchester United.
Much could depend on whether Chelsea draft in another striker themselves in January, otherwise all interested parties may have to wait until the end of the season if they are to have a chance of doing a deal.
Giroud, who is a World Cup winner with France, will not be short of options and it remains to be seen whether he would be interested in a move north, if Chelsea did let him go.