The 25-year-old is the older brother of Juventus superstar Paul Pogba, and plays for French side Saint-Etienne.
Florentin Pogba has been at the club since 2012, and despite playing for France at under-20 level, is now a senior international for Guinea.
Pogba plays at centre-back and is under contract until 2018. He has made 18 appearances this season, including six in the Europa League.
Both Villa and Norwich are looking to strengthen in order to survive in the Premier League, but Remi Garde's side's prospects are more bleak, currently 11 points away from safety, which may harm their chances of signing the defender.
The Canaries are in 15th place, currently six points ahead of the relegation zone, having amassed 23 points so far this season.
Pogba started out as a youth with Spanish side Celta Vigo, before signing with French club Sedan in 2009.