The 22-year-old midfielder has signed a five year deal with the Midlands club after having a medical on Friday.
Veretout becomes Tim Sherwood’s third signing this week, joining Jordan Ayew and Jose Angel Crespo.
The France U21 international has been a regular in the Nantes side for the past four seasons.
And Sherwood said he is delighted the Frenchman has chosen Villa.
“He is only 22-years-old but he is vastly experienced for someone that age,” he told the club’s website. “He has played over 140 games in France and is a really exciting young player.
“I know he is very highly rated over in France so we are delighted to get this deal over the line.”
Veretout represented his country at U18 level and won the U20 World Cup in 2013.
Earlier this week, Nantes president Waldemar Kita said he agreed to sell the player to Villa despite a better offer from Leicester.
He told the newspaper L’Equipe: “I am not going to go against the boy. He is a boy who has never done us wrong, who played the game right until the end, contrary to others, who did not sulk.”