Goal say their initial offer exceeds £35million and that they are keen to sign him after tracking him since his days as a youngster with Paris Saint-Germain.
It is also claimed the move has been backed by his fellow Frenchman and Red Devils legend Eric Cantona who currently works as an ambassador for United.
Dembele, 22, has impressed in France after joining Lyon from Celtic at the start of last season.
Despite only making 20 Ligue 1 starts in 2018-19, the striker bagged 15 goals and has proved he can shine in one of the big five European leagues.
While the two clubs are said to be miles apart in terms of their valuation of the Belgian striker, reports have claimed they are working on a swap deal that could involve either Ivan Perisic or Milan Skriniar moving to Old Trafford.
Meanwhile, Dembele could become the first signing of Solskjaer’s summer rebuild
The Norwegian is looking to bring in a number of fresh faces after a disappointing end to the season that saw them finish sixth in the Premier League and fail to qualify for the Champions League.
Express Sport understands Solskjaer is keen to bring in midfielder Declan Rice from West Ham and Borussia Dortmund winger Jadon Sancho.
He is desperate for reinforcements in almost everywhere, although a new centre-back remains a priority.
Leicester star Harry Maguire continues to be linked with a switch as well as Napoli’s Kalidou Koulibaly.
Further forward, Lille sensation Nicolas Pepe is also said to be high on their list of targets after a stunning season that saw him bag 22 Ligue 1 goals.