The 30-year-old started just eight Premier League games last season and United boss Erik ten Hag is open to letting Maguire leave in the transfer window.
When asked last week about Maguire's lack of game time Ten Hag said: "I'm happy he's here... it's also a decision he has to make [because] nobody would be happy with this situation."
Maguire is under contract at Old Trafford until 2025 with an option to extend by a further year and the centre-back has previously been determined to stay and fight for his place.
However, with United willing to let Maguire leave the club, it remains to be seen whether a firm offer from Spurs, West Ham or another club would be enough to convince him to attempt to revive his fortunes elsewhere.
Although Postecoglou does not officially start work at the club until July 1, the 57-year-old will help form the club's transfer plans this summer and Maguire's name is expected to be discussed.
Maguire's wage is a potential stumbling block for the north London club. Sources have told ESPN he earns in the region of £200,000-a-week but that figure is set to rise after United qualified for next season's Champions League.
Harry Kane is Tottenham's top earner on the same amount and Maguire would likely have to take a pay cut to join Spurs, as would be the case with West Ham, who enquired about the England international in January.
Maguire was unwilling to move to east London at the time and West Ham's interest was driven by manager David Moyes, whose own future is unclear ahead of Wednesday's Europa Conference League final against Fiorentina.
If Moyes stays on, the club are likely to pursue a deal for Maguire. England boss Gareth Southgate warned Maguire last month that his lack of first-team football at United is "not a situation that can continue for ever" if he wants to keep his international spot.
Southgate has remained loyal to Maguire throughout his time in charge of England, starting the defender in all five of the team's matches at the World Cup in Qatar but his place will likely come under renewed threat ahead of Euro 2024.