The 26-year-old defender has struggled to stay fit over the last two seasons, missing nearly 230 days over the past year-and-a-half.
The Sun says the Ivory Coast international – whose terms runs until June 2022 – has recently returned and put in some solid performances.
And Bailly is now in early discussions over a new contract, thought to be around three-years in length.
But the ex-Villarreal ace is facing an uncertain future at Old Trafford if he does not sign fresh terms by the summer.
United could then be prepared to listen to offers for the centre-back, who has become a favourite among supporters.