The 28-year-old is expected to leave Old Trafford this summer following the big-money arrival of Lisandro Martinez from Ajax.
Bailly is currently one of eight central defenders at the club, with Martinez, captain Harry Maguire, Raphael Varane and Victor Lindelof all seemingly ahead of him in the pecking order.
Man United are believed to be willing to sell Bailly for less than £10m this summer, which appears to have alerted a number of European clubs.
A recent report claims that newly-promoted Fulham have stepped up their interest in the Ivory Coast international, who has two years remaining on his contract.
However, Calciomercato states that Roma are 'closing' in on the signing of Bailly after holding talks with the Red Devils.
The report adds that Giallorossi are discussing a possible loan deal with an option to sign the defender permanently next summer for around €8m (£6.7m).
Roma manager Jose Mourinho was the man who signed Bailly during his time at Man United, with the centre-back joining from Villarreal for around £30m in 2016.
Bailly has since made a total of 113 appearances for the Red Devils across all competitions, but he struggled for game time last season.
The Ivorian featured only seven times in the 2021-22 campaign under both Ole Gunnar Solskjaer and interim boss Ralf Rangnick and was handed only three starts in the Premier League.
Bailly recently hinted at a potential move to Roma after liking an Instagram post from journalist Fabrizio Romano unveiling the arrival of Paulo Dybala.
Dybala is one of four players to arrive at the Stadio Olimpico this summer along with Zeki Celik, Mile Svilar and Nemanja Matic, the latter of whom has joined on a free transfer from Man United upon the expiration of his contract.
Bailly is not the only United player who could be joining Matic in the Italian capital, as a recent report claims that Mourinho is also interested in reuniting with Victor Lindelof.
The 28-year-old, who was another defender signed by Mourinho at Old Trafford, was allegedly mentioned in negotiations between United and Roma while talks were held over a deal for his teammate Bailly.
While Lindelof is believed to be part of Erik ten Hag's plans for the new season, the futures of fellow defenders Aaron Wan-Bissaka, Phil Jones, Axel Tuanzebe and Brandon Williams are uncertain.
Meanwhile, Alex Telles has completed a season-long loan move to La Liga outfit Sevilla, with the left-back leaving Old Trafford on a temporary basis after spending two years in the senior side.