Mundo Deportivo understands that the Premier League title hopefuls have submitted a bid worth €40m (£36.1m), but are not the only English club looking to acquire the 25-year-old. Cross-city rivals Manchester City were believed to have made their own approach last month, while Chelsea are also now said to be chasing his signature.
United have apparently made Roberto aware of their interest, although Barcelona insist that he is not for sale and the player is reportedly eager to remain at the Nou Camp.
A graduate of the Blaugrana's prolific La Masia academy, Roberto featured predominantly as a right-back during Luis Enrique's final season at the helm in 2016-17 following the departure of Dani Alves to Juventus.
Emerging ahead of Aleix Vidal as the first-choice option, he went onto make 47 appearances across all competitions.
Despite Roberto impressing in that previously unfamiliar role, Barcelona made the addition of a new right-sided defender a priority following the appointment of new manager Ernesto Valverde.
After failing with their well-documented attempts to secure a return to Catalonia for Arsenal's Hector Bellerin, the 24-time Spanish champions, also linked with the likes of Joan Cancelo, Cedric Soares and Alvaro Odriozola, eventually signed Portuguese international Nelson Semedo from Benfica in a deal initially valued at €30m.
Roberto will now presumably revert back to his preferred midfield role, where he will compete with the likes of Andres Iniesta, Sergio Busquets and Ivan Rakitic for a starting berth. Competition for places in that area of the pitch is likely to increase before the transfer deadline as Barcelona continue to be linked with moves for Liverpool's Philippe Coutinho and Guangzhou Evergrande's Paulinho, among others.
United have made three signings this summer to date, with defender Victor Lindelof and striker Romelu Lukaku recently joined at Old Trafford by former Chelsea midfielder Nemanja Matic. Jose Mourinho is known to be targeting the addition of four new players before the end of the window, with a wide player now top of his wishlist as speculation regarding Inter Milan winger Ivan Perisic persists.
Chelsea, meanwhile, have added Willy Caballero, Antonio Rudiger, Tiemoue Bakayoko and Alvaro Morata to their title-winning group over recent weeks. However, their bench for Sunday's (6 August) Community Shield defeat to Arsenal at Wembley appeared notably thin following the departures of several second-tier players and a frustrated Antonio Conte is targeting further reinforcements ahead of the club's imminent return to the Champions League.
With Matic now gone and Bakayoko still recovering from surgery on a minor knee injury, the Italian seems likely to prioritise the acquisition of another new midfielder.