The England international made the move to the Giallorossi on a temporary deal last summer and he has since been one of the league's best-performing centre-backs.
As such, the likes of
have recently been linked with the defender's services but Corriere della Serra claims that the Giallorossi are keen to beat them to the player's signature.
United were initially said to value Smalling at around £12.5m but they have supposedly doubled the asking price following the Englishman's success in the Italian top-flight this term.
Smalling is prepared to take a paycut to push through a permanent move to the Stadio Olimpico but it is unclear whether the Serie A outfit will meet United's valuation for the ex-Fulham graduate.
Smalling has more than two years remaining on his existing deal at United. His performances for the Giallorossi have recently led to suggestions that he could be recalled to the England national side.