The Daily Star says United have entered the race to sign Calafiori.
The Red Devils have watched the 18-year-old left back playing for Roma's youth team several times this season.
And the Old Trafford giants have now sounded out his agent Mino Raiola about a possible transfer.
Calafiori, who is under contract with Roma until 2022, is already wanted by Paris Saint-Germain.
And he has been offered a bumper new contract in a bid to convince him to stay, with Roma keen to bring him into the first-team squad next season.
Technical staff at the club also believe that while the Italy Under 19 international is a top prospect, it's too early for him to move abroad. But it is understood Roma would be willing sellers at the right price and are hoping to encourage an auction.