Smalling, who has been at Old Trafford since 2010, spent the 2019-20 campaign on loan to the Serie A club and impressed in the Italian capital, making 37 appearances in all competitions.
Corriere dello Sport says Roma have been pursuing a centre back all summer and Smalling, 30, remains their number one target. Their latest offer of £10.9m (€12m) plus £2.7m (€3m) in add-ons is currently being considered.
Roma's current offer is reportedly valid for 48 hours - the international transfer window closes on Monday.
United are aware of the need to sell the English defender this summer, instead of receiving a smaller fee for him next year.