The Sporting Lisbon left-back was the subject of United scouts' attention earlier this month, with Old Trafford boss David Moyes reportedly on the lookout for a replacement for Patrice Evra.
The French international is expected to leave United at the end of the season as his contract expires.
And United officials have reportedly been considering a summer move for Sporting's left-back Rojo, who is equally comfortable playing in the centre of defence.
The 23-year-old Argentina international has 19 caps for his country, and is hopeful of making the World Cup squad for the tournament in Brazil this summer.
And his impressive performances in Portugal have attracted attention from clubs in Italy as well as England.
According to A Bola, Roma, Inter Milan and Napoli are the three teams believed to be tracking Rojo, with Roma in particular keen on the defender.
Walter Sabatini is reportedly a big fan of the Argentine's versatility, with Roma thought to be willing to bid up to €8m to land him before the World Cup.