The Ivory Coast defender, who was former United manager Jose Mourinho's first signing at Old Trafford with a £30 million transfer from Villarreal in June 2016, has been in talks over a new deal to replace his contract, which expires at the end of next season.
But sources have told ESPN that the 27-year-old has become increasingly frustrated with being overlooked by Solskjaer, who favours Sweden international Victor Lindelof as defensive partner for club captain Harry Maguire at centre-half.
Bailly has made just 100 appearances in all competitions for United since arriving five years ago, with the former Espanyol defender's time at the club disrupted by a series of lengthy injury lay-offs.
He has appeared in only 15 of United's 50 games so far this season and Bailly has been unavailable for the club's last three games after testing positive for COVID-19 while on international duty last month.
Sources have said that Bailly could return to contention in time for Thursday's Europa League quarterfinal second-leg against Granada at Old Trafford, but after failing to convince Solskjaer that he should be Maguire's regular partner, he is reluctant to commit to a longer contract at United and face more time on the substitutes' bench next season.
United would risk losing Bailly as a free agent next summer if he remains at the club without extending his contract and, with interest from clubs in Spain and Italy, a move is likely to suit both parties at the end of this season.