The Englishman moved to the Italian capital over the summer on a season-long loan deal following the arrival of Harry Maguire.
His performances in Serie A have not gone unnoticed, with the Englishman cementing himself as first-choice under Paulo Fonseca.
According to the Daily Star, Roma submitted a bid of £13m for the defender that United have since rejected.
Smalling, who moved to Old Trafford in 2011, is said to be valued at around £15m by his parent club, who are open to the idea of selling.
Having enjoyed such a successful start to life in Italy, Smalling is said to be keen on remaining at Roma on a permanent basis and has also attracted interest from other Serie A sides.