The 26-year-old has been with Marseille since January 2017 and has been linked with a succession of Premier League clubs, including West Ham.
Villa have now begun talks with the French side and a deal could progress in this month's transfer window.
Marseille are prepared to sell in order to fund their own January signings.
Sanson - whose value has dropped from around £25m - would bolster Villa's squad in midfield, with Conor Hourihane being linked with a move away from Villa Park.
Marseille manager Andre Villas-Boas said at the weekend: "The transfer window is open. Sanson is a player who is highly rated.
"He is a boy who interests English teams because of his characteristics and it is possible something could happen."