Sky Sports says Dembele wants to move to Old Trafford.
Dembele will be allowed to leave Lyon at the end of the season and he wants to return to England where he started his career at Fulham.
Dembele has scored 13 times this season and injuries to key players meant Lyon were unwilling to let him leave in January.
They also wanted to keep him because they are still in the Champions League and face Juventus next Wednesday.
Lyon president Jean-Michel Aulas will let Dembele leave in the summer but he will have to settle for less than his €100m (£83m) asking price.