The deal also includes buy-back and sell-on clauses and the fee is expected to rise £21.7 million.
The young Dutchman has made eight appearances for United this season, but only one start, and has featured for just eight minutes since October under Jose Mourinho.
His days at the club seemed numbered when Jose took over, and the manager's preference for players to fit his own identikit was always going to make Depay an outcast.
Memphis was the top scorer for PSV in the Eredivisie in the 2014-15 and that form persuaded Van Gaal to bring him to United for £31 million.
Highly regarded as one of the hottest young properties in Europe, Memphis future at United seemed destined to be a bright one as he started strongly and dovetailed well with Luke Shaw.
The two had begun to form a potent partnership down the left and two stunning goals - albeit against the lesser lights of Brugge - demonstrated the 21 year old's potential before Shaw broke his leg in Eindhoven.
Memphis never regained his previous form and - despite flashes of brilliance against Midtjylland, Watford and Arsenal, he fell out of favour and was dropped by Van Gaal.
There's no doubting his talent but his attitude and willingness to work for the cause were often called into question.
The most obvious example of this lazy approach came at Chelsea when, after he had come on as sub, his lack of tracking back led to their breakaway down his side of the pitch and Diego Costa scored.
Tellingly, he becomes the second big-money, high profile Van Gaal signing to be moved on.
Morgan Schneiderlin was also hampered by lack of game time under Mourinho and signed for Everton in a £26 million move last week.
Angel Di Maria became our record signing when he signed for £59.7 million but he also failed to live up to expectation and left for PSG after just one season in Manchester.
Bastian Schweinsteiger has been heavily linked with a move away and Luke Shaw's fitness and inability to adapt to Jose's new United style has also left his future at the club an uncertain one.
Memphis becomes the second Red to move to Lyon in recent seasons after Van Gaal sold former full-back Rafael to the French outfit in 2015.
They currently sit fourth, eight points off the Champions League qualifying places.