The Ivory Coast international was a regular during his first season at the club, making 38 appearances in all competitions.
However, Bailly made just 13 league appearances last term and was an unused substitute in the FA Cup final defeat to Chelsea in May.
Mourinho was said to be annoyed by Baillys decision to pull out of a game against West Brom in April and the defender was then left out of matchday squads against Spurs and Arsenal.
United have been heavily linked with a move for Tottenham star Toby Alderweireld and the Belgians arrival would push Bailly further down the pecking order.
According to the Manchester Evening News, Bailly is very happy at the Theatre of Dreams but is unwilling to sit on the bench.
Bailly would not be short of suitors should he decide to leave Old Trafford but United would demand an increase on the £30.7m they paid for him in 2016.
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United hope to complete a deal for Brazil midfielder Fred during the 25-year-olds stay in England for a friendly with the national team against Costa Rica.
The Shakhtar Donetsk star is set to cost up to £53m, eating into Mourinhos £150m budget this summer.