Depay could make his debut in Sunday's Ligue 1 clash at home to Marseille, AFP reported.
No financial details were released, but Lyon president Jean-Michel Aulas, asked by RMC Sport radio, suggested the deal was worth €16.5 million ($17.5 million) plus "eight million in bonuses".
That would make Depay, capped 27 times at international level, Lyon's biggest signing since the club-record purchase of Yoann Gourcuff from Bordeaux for €22 million in 2010.
"Hungry to play for OL. I feel strong and ready to rise to the challenge #TeamOL," Depay wrote on his Twitter account.
"Thanks for all the memories during my time as a Red Devil. I will never forget all your support at @ManUtd. #RedDevils #ManUtd"
The 22-year-old joined United from PSV Eindhoven in June 2015, but the winger had a disappointing first season under compatriot Louis van Gaal and had made just one start in all competitions since Jose Mourinho became manager last July.
The deal reportedly includes buyback and sell-on clauses, and Mourinho refused to rule out an eventual return to Old Trafford for the Dutchman.
"He didn't succeed in this 18 months, I would say, but he is very young so I think it is important for the club to keep control of this talent," said the Portuguese.
"And we all wish that he plays very, very well at Olympique Lyon and why not to come back because everyone here likes him."
Depay, who scored seven goals in 53 United appearances, becomes the second player to leave Old Trafford this month after France international midfielder Morgan Schneiderlin joined Everton.