I took a brief look at Alizée's IG profile and apparently she doesn't follow any football-related account...
We know she used to play football with her brother and I might have read on this forum that her father is a fan of Olympique de Marseille. Also, there are some photos that shows her interest in the national team.
And most importantly, she played as playmaker for Fc Ajaccio, leading the team to Champions League! Unfortunately she retired after that and the team was relegated to the second division.
Ok, maybe... Um... No.
Anyway, I presume she has always been a football fan.
But I don't know whether she watches every match of her club or she likes watching something every now and then.
If you ask me, I think she is an occasional fan that probably prefers
playing to
watching .
As for Arsenal, she may know the club. I think it is the most notable foreign club for a French football fan, because, as you said, French players have had an important role in its history. Giroud also played for Arsenal some years ago and she knows him because he won the World cup.
Another Premier League team which has had many French players is Newcastle FC but I don't think she knows it.
I remember there is a photo where Alizée is holding a ball and, if I'm not mistaken, it was signed by the players of the national team. I think it is from 2013 or so. Later I'll see if I can find it.