The 25-year-old has impressed at this summer's World Cup and is now on the rader of Europe's top clubs, including the Red Devils, per Team Talk.
It is believed that Kone wants to test himself in the Premier League, and that the Serie A side would be happy to sell for the right price.
United have already agreed deals for two midfielders this summer, the Brazilian pairing of Ederson and Andrey Santos for a combined £85 million, and they could be set to bring a third to Old Trafford ahead of next season.
Kone, recognised as both a box-to-box and defensive midfielder, made 37 appearances for Roma last season, scoring twice and registering two assists.
The midfielder as improved his reputation further after impressing in the centre of the park for Didier Deschamps' France side, and may well have earned himself a big move this summer.