It is thought the fee is £16m rising to £21.7m, with the add-ons including Lyon qualifying for the Champions League and Depay getting a new contract.
United have also agreed buy-back and sell-on clauses.
Netherlands forward Depay, 22, has scored seven goals in 53 appearances since joining United in a £31m move from PSV Eindhoven in May 2015. The initial fee was about £25m.
The deal with PSV is thought to have included a number of add-ons which have not been met.
Lyon are fourth in Ligue 1, 11 points behind leaders Monaco and eight points behind PSG in third, the closest Champions League qualification place.
This season Depay has made eight appearances for United, but has featured for eight minutes since the end of October.
He was the Dutch Eredivisie's top scorer in 2014-15 and was brought to United under then-manager Louis van Gaal, who had given him his Netherlands debut.
He becomes the latest high-profile Van Gaal signing to be sold on.
Argentina attacking midfielder Angel di Maria signed for a then-British record fee of £59.7m in August 2014, but was sold to Paris St-Germain for £44.3m a year later.
Morgan Schneiderlin, who signed from Southampton for £25m, was sold to Everton on 12 January for a fee rising to £24m.
Former Germany captain Bastian Schweinsteiger was also brought to Old Trafford by Van Gaal, joining from Bayern Munich for a reported £14.4m.
However, the 32-year-old has played just 16 minutes under current boss Jose Mourinho this term.
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Manchester United have agreed a deal to sell Memphis Depay to French club Lyon.