The 18-year-old, who has been linked with a move to Chelsea, remains under contract at Old Trafford until 2027, but the club's part-owners, Ineos, are also keen to extend his stay.
Mainoo's importance to United has grown this season, making him a priority for the club's long-term plans.
His potential exit would be a major blow, especially as the club looks to stabilise under Amorim following a disappointing 2-0 defeat to Crystal Palace at the weekend.
Meanwhile, United made two signings in the January transfer window, with left-back Patrick Dorgu arriving from Lecce and young defender Ayden Heaven joining from Arsenal.
Another addition, Paraguayan talent Diego Leon, has also signed from Cerro Porteño but will not join the club until he turns 18 due to post-Brexit regulations.
Looking ahead, United will aim to bounce back from their latest league setback when they face Leicester City in the FA Cup at Old Trafford on Friday evening.
Leicester's manager, Ruud van Nistelrooy, briefly took charge as an interim boss earlier this season before returning to management with the Foxes.
Tyrell Malacia admits he cannot wait to be "finally playing good football again" after finalising a move to PSV Eindhoven as one of two beat-the-deadline deals confirmed by Manchester United.
Manchester United staff are reportedly relieved that Alejandro Garnacho remains at Old Trafford after the club rejected multiple approaches for the young winger.
Marcus Rashford's tenure at Manchester United appears to have ended definitively, as reports suggest he is 'extremely unlikely' to play for the Red Devils again, even if head coach Ruben Amorim were to depart.
In a disheartening Premier League display at Old Trafford, Manchester United fell to a 2-0 defeat against Crystal Palace, with Lisandro Martinez's injury adding to the gloom.
