Lyon paid 16 million euros ($17 million) for the 22-year-old Netherlands international.
Memphis scored seven goals in 53 appearances for United after moving from PSV Eindhoven for 25 million pounds ($30 million) in May, 2015 but he has played only eight minutes since the end of October.
"He was a fantastic professional," United manager Jose Mourinho told a news conference. "If someone thinks it didn't work because he wasn't a great professional, that's totally wrong."
Memphis was the leading scorer in the Dutch top flight in 2014-15 and was signed by former United manager Louis van Gaal.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
But under Mourinho he has fallen down the pecking order among the club's wide players, behind the likes of Henrikh Mkhitaryan, Juan Mata, Anthony Martial and Jesse Lingard.
"When Mr Van Gaal bought him, he did well," Mourinho said, adding that United had inserted a buy-back clause into the deal with Lyon.
"He didn't succeed in these 18 months, but he's very young. It's important for the club to keep control of this talent and we wish that he plays very well at Lyon. Why not come back here, as everybody likes him."
Lyon are fourth in French Ligue 1, 11 points behind leaders Monaco.
(Editing by Ed Osmond)
