Memphis joined United from PSV in the summer of 2015 but struggled to hold down a regular place under both Louis van Gaal and Jose Mourinho.
He leaves Old Trafford just 18 months into his four-year contract having scored just seven goals in 56 appearances for the 20-time English champions.
He has made just eight substitute appearances for the Red Devils this season, with his Premier League outings amounting to just 20 minutes of action, and becomes the second high-profile departure from United in the January transfer window following Morgan Schneiderlin's move to Everton.
NOUS SOMMES L'OLYMPIQUE LYONNAIS pic.twitter.com/msXGylCVjg— Memphis (@Memphis) January 20, 2017
Memphis will wear the No.9 shirt for his new club and could make his debut against Marseille on Sunday.