The club have also confirmed the departure of Ruben Vinagre for Sporting Lisbon in a deal which should see them net £8.5m.
Ait-Nouri comes in on a five-year contract – with the option of a sixth – having impressed on loan at Molineux last season.
Angers hold a 50 per cent sell-on clause which Wolves can buy out a later date.
Technical director Scott Sellars said: "We think Rayan is a great signing for the football club.
"He proved last season that he’s a very talented player, with lots of potential.
"He’s a player who is only going to get better. He’s a good boy, who works hard, and settled in well here.
“We’ve used the loan before buying approach well in the past, with people like Jonny (Castro Otto) and Raul (Jimenez).
"I think it’s a great way for both parties to get to know each other, to see if they can adapt and enjoy English football, and we get a chance to have a look first, and see if a deal can be done.
“In Rayan’s case, he got a lot of minutes because both Jonny and (Fernando) Marcal suffered injury problems last season, so he had lots of games in the Premier League to show his quality.
“At his age, he’s got a lot to learn still, a lot of work on, and I’m sure with us he’ll develop."
Ait-Nouri is Wolves' third signing of the summer, after Yerson Mosquera and Francisco Trincao.
Vinagre, meanwhile, joins Sporting on an initial season-long loan which is expected to become permanent.
It is understood there are appearance-based clauses that would automatically trigger an £8.5m move to the Portuguese champions, officially ending his Molineux tenure.
Vinagre has played 70 times since joining Wolves in 2017 and had loan spells at both Olympiacos and Famalicao last season.