The 20-year-old, who is regarded as one of the best young full-backs in France, made his senior debut for Lens in 2009.
"I would like to perform at the top level and play against top opponents every week. I am enjoying my international career so far. I love the national team, they have been great for me," Aurier told reporters in France.
"Everton are a top team and if they show interest in you it means a lot. I am just trying to stay relaxed about things. Marseille are a top club in France and when they show interest you cannot ignore it, you have to listen."
Aurier joined Toulouse in 2012.