Maguire is likely to have to settle for a role on the United bench this coming season with Lisandro Martinez and Rafael Varane as the first choice pairing.
The Hammers have been touted as a potential destination and talkSPORT understands that David Moyes is a fan.
talkSPORT brought you the news in June that Maguire was willing to fight for his place at the Red Devils, something that remains unchanged at this stage.
Any approach from West Ham is still a long way off, but former Chelsea striker Tony Cascarino has urged the 29-year-old to follow Chris Smalling's example.
He told Weekend Sports Breakfast: "Chris Smalling left and went to Roma and West Ham would be a great move for Harry if it came about.
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"Chris Smalling has done brilliantly well and has had some great seasons in Italy. Harry would get his confidence back and having a manager who picks you is important."
With Euro 2024 also on the horizon, Cascarino has suggested that Maguire should want to move so he can guarantee his place and form going into that tournament.
He added: "He has had a difficult season. The Brentford game away where he had a poor game and they defended terribly.
"He found his way back, but it has been made quite clear that he's not going to be in the first XI.
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"If you're a centre-half going into a tournament next summer you want to be playing and you should be playing.
"I do feel that an opportunity [at West Ham] and his role would be slightly different under David Moyes where he wouldn't have to play so much football at the back with the ball.
"Harry is comfortable, but the level of expectation that Ten Hag has at Manchester United [is different], you have seen that he's buying a goalkeeper in [Andre] Onana who is ultimately a footballer.
"Harry lost a little bit of belief in how he was over playing and he's not as gifted as John Stones with the ball at his feet.
"I don't think Harry Maguire would be asked to do that at West Ham."