The Gunners are said to have been tracking the 24-year-old, who moved to St Mary's from Strasbourg in 2008, for some months.
According to The Metro, Schneiderlin told French media: "It is my dream to one day play at a great club. Arsenal? I have a three-year contract, so it will not be easy. It's my dream to one day play at a great club though, so why not?"
The Saints will take on Arsenal's North London rivals Tottenham Hotspur on Sunday.