The 28-year-old is expected to leave the Red Devils this summer, with the big-money arrival of Lisandro Martinez from Ajax pushing the Ivorian further down the pecking order.
Bailly is currently one of eight central defenders on the books at Old Trafford, with Martinez, captain Harry Maguire, Raphael Varane and Victor Lindelof believed to be the top-four options in Erik ten Hag's plans for the new season.
A recent report claimed that Jose Mourinho's Roma were leading the likes of Fulham and Sevilla in the race for Bailly's signature.
During his time in charge at Man United, Mourinho signed the defender from Villarreal for around £30m in the summer of 2016.
However, French news outlet L'Equipe reports that Roma will face competition from Ligue 1 outfit Monaco to sign Bailly this summer.
The report adds that Monaco are exploring the possibility of taking the Ivory Coast international on loan with an option to buy next summer.
It is thought that United would prefer to sell Bailly this summer to generate funds for potential incomings, and would be willing to accept offers less than £10m.
A report from ESPN, meanwhile, claims that Bailly is looking to push through an exit from Old Trafford before the September 1 deadline.
Monaco's potential swoop for Bailly will not hinder their pursuit of Chelsea defender Malang Sarr, who is reportedly edging closer to joining the club on loan with an obligation to buy after completing his medical on Monday.
Monaco have allegedly agreed to pay a loan fee of £600,000 for the 23-year-old and will sign the Frenchman permanently for around £13m if he achieves a number of performance-related bonuses.
Should Monaco finalise deals for both Bailly and Sarr, the defensive duo would become the club's fourth and fifth signings of the summer window after the arrivals of Takumi Minamino, Breel Embolo and Thomas Didillon.
Bailly is not the only Man United defender who faces an uncertain future at Old Trafford, as the likes of Aaron Wan-Bissaka, Brandon Williams, Phil Jones and Axel Tuanzebe have all been linked with moves away this summer.
Wan-Bissaka, who has fallen behind fellow right-back Diogo Dalot in the pecking order, is believed to be attracting interest from former club Crystal Palace.
It is thought that United would be open to the prospect of either loaning or selling Williams this summer, while both Jones and Tuanzebe are deemed surplus to requirements under Ten Hag and have less than 12 months remaining on their contracts.