Dutch international Depay, who joined United from PSV in 2015, has struggled for regular first team football at Old Trafford this season under Jose Mourinho, with a transfer out of the club being touted for weeks.
The 22-year-old has only made eight appearances for the Red Devils this season, the last of which came against Feyenoord in the Europa League in November.
The winger, who has played 53 times for United and netted seven goals, would become the second high-profile signing to leave Old Trafford in this transfer window after midfielder Morgan Schneiderlin joined Everton recently.
The Toffees' manager Ronald Koeman also expressed an interest in buying Depay but it seems as if Lyon has beaten all other clubs to his signature.